Bloggers Need God
Such was the informal title I had given to the growing list of links on my computer’s calendar, and I thought it was sort of catchy so I used it here.
I recognize this particular Sunday post might sound more aggressive than my usual fare, like I’m now being more a stereotypical American evangelical who wants to ram my worldview down everyone’s throat. All I can say is that I used to be a “devout atheist” (my term at the time), so I don’t think the people I’m about to criticize are stupid. I used to hold views like theirs, in several of the examples. Yet now that I believe in God and am no longer trapped in the materialist mindset, it is amusing and alarming to me how such bright people (including my former self) could have so easily fallen for such fallacies. (And yes, I realize the immediate reply will be, “I know you are but what am I?”)
The point of this post is to go through several examples of the bloggers I frequently read, where they make very simple errors that are glaringly obvious to someone who believes in a God of the popular monotheistic traditions. In most of the examples (not all) even an atheist should be able to spot the error, but the point is that there wouldn’t have even been the slip-up had the person reflected on the nature and existence of God. It was the blogger’s agnosticism (or at least, lack of believing in anything like the Judeo-Christian / Muslim God) that left him vulnerable to the mistake.
Two caveats: (1) If I am wrong in assuming that each of these bloggers doesn’t believe in this type of God, then my apologies. I will correct the post if anyone shows me otherwise. (2) I’m going to move from my weaker examples to the stronger ones, so in the beginning even those of you who agree with me might not think it’s a big deal.
To get the ball rolling we’ll start with Bryan Caplan’s recent post on a former slave who wanted (perhaps apocryphally) back wages from his former master before he’d work for him again. Bryan quoted the former slave saying: “Surely there will be a day of reckoning for those who defraud the laborer of his hire.” Then Bryan commented: “Too bad the last sentence turned out to be wrong. Life is not fair.”
Now here, I’m mostly nitpicking. If Bryan had known more about the Bible, he would have understood that this was a Scriptural reference. So this guy is saying to his former master, “You had better repent of your sins before meeting your Maker.”
But the thing is, Bryan’s statement is actually silly, even if we don’t believe in the afterlife. Does Bryan, the tenured college professor, really think he needs to inform a former plantation slave that life isn’t fair? So my point is, whether you believe in the afterlife or not, Bryan’s statement makes little sense. It’s not that Bryan observed that there’s no afterlife, and so now he empirically can conclude, “Too bad, that guy’s hypothesis was falsified.”
My next blogger is Karl Smith, who is the James Joyce of economics blogging. In this post, Karl makes an analogy of the economy as a giant forest, and talks about why we should try to save the trees. He writes:
[W]e shouldn’t sit by while a new virus sweeps through and destroys the trees we love. In perhaps the grandest sense we could say, that yes these trees may die but we don’t worry because eventually they will be replaced by other trees in a never ending circle of life.
This is very true. But, we care for and love these trees — and that matters. On a deepest level it matters because our emotions are the ultimate source of value. At their core the trees are just another set of molecules. They are beautiful because they are beautiful to us.
It’s the part I put in bold that concerns me. This is materialism in all its beauty / ugliness, depending on your value system. First, note that this is a completely arbitrary statement on Karl’s part. Does the Pythagorean Theorem exist? It doesn’t consist of molecules. Indeed, do molecules exist? Physicists will say that they aren’t really “solid” things either; they’re mostly empty space. If you really push it, according to cutting edge theories matter itself becomes more and more like an idea, rather than that “hard stuff” that’s “really” “out there” as opposed to the “not as real” stuff that’s in our minds.
If you want to be really basic about it, the notion of a physical universe is a theory that we use to explain the more fundamental sensory data that we experience. After all, we might all be in The Matrix.
The true irony in Karl’s statement about the trees being “at their core” a bunch of molecules, is that in another post he writes:
On video I have a tendency to smile and laugh a lot. I am also a generally happy-go-lucky type of person. This combined with my sloughing off of long term issues has people often mistake me for a Pollyanna. That is someone who thinks everything will be ok.
Ironically – or not – I believe the exact opposite. Everything will definitely not be ok. I often tell my students: if you ever find yourself worried sick about whether or not things are going to turn work out, don’t worry, things are definitely not going to work. Everything is going to go horribly, horribly badly.
This is the essence of life. We are not forever. Our institutions are not forever.
Indeed, in a relatively short time, we and all the things that matter deeply to us will be annihilated. They will not exist at all and they will never come back. Not at least as we would think of such. There will be no faint hint of them in the background of the universe or spirit occupying another plane.
All things we care about are at their heart information – particular arrangements of the building blocks of reality – and entropy eats information. Everything we care about will be gone.
Yikes! Talk about a blogger needing God! Karl, please entertain the idea that there reputedly was a man who said many things that you would agree are very wise and good, and that this same man reputedly said paradise awaits us if we don’t reject it.
Beyond the tremendous burden of walking around with that worldview, Karl’s statements are (again) arbitrary and unscientific. He says these things won’t be in a “spirit occupying another plane,” but modern science doesn’t tell us that. And to tell people that “in a relatively short period of time” entropy will engulf everything we care about is pretty close to demonstrably false. I mean, relative to what?! Karl is saying humanity will necessarily be extinguished, when every possible reference point is also extinguished. Short of eternity, what could be longer than the maximum age allowed by the laws of physics?
Of course, the other problem is that Karl is here betraying the materialism from his other post. Information is itself not a physical thing. For humans to perceive it with their sense organs, it must be instantiated somehow, I grant you. But the information itself is more than the physical components that represent it. (Read Gene Callahan’s great post on these themes.)
Now we’re moving on to a really fun one. In this post, Scott Sumner was criticizing the Rothbardian view of the Great Depression. Sumner was arguing that the Fed couldn’t possibly have caused an inflationary boom in the 1920s. In the comments I asked him to clarify one of his arguments that amazed me, and he said:
I’ve shown there was no inflation as the term was defined at the time. I’ve shown that there was no alternative non-inflationary policy as understood by policymakers at the time, including those in the 1920s who claimed the Fed was too inflationary. It makes no sense to argue things were inflationary because M2 went up, if M2 didn’t exist. There are no policy implications. M2 was an idea invented much later.
One doesn’t have to be a Christian to see the non sequitur. As I wrote here, “I wonder how Sumner explains the massive deaths during the bubonic plague? Did doctors even know what bacteria were back then?”
I am naive. I expected Scott to say something like, “Wow, I don’t know how that one slipped through my keyboard. Sorry guys, that was silly. But I still don’t think the Fed created a ‘bubble’ in the stock market, the way the Austrians claim.” Yet to my knowledge, Scott offered no such retraction. I don’t have the links, but Scott’s views on the nature of reality are downright freaky. I am not putting words in his mouth, he has said (paraphrasing) that phenomena exist when the top experts in that field agree that such existence would prove useful. If you agree with me that this type of view is freaky, I note that–whatever else you want to say about monotheism in the popular traditions–it blows up that sort of view really quick.
And last we come to a very religious atheist, Steve Landsburg. In a passage about Godel’s work that truly fills me with brotherly love, Steve writes:
The code is cleverly constructed so that there’s a statement in pure arithmetic (say, for illustration, that it’s the statement “every even number is the sum of two primes”) that corresponds to the English sentence “The statement that every even number is the sum of two primes cannot be proven.” These statements are either both false, in which case it’s possible to prove a false statement, which we believe (and hope to God!) is not the case — or they’re both true, in which case we’ve found a true statement in pure arithmetic that can’t be proven.
You’re right, Steve: I don’t think arithmetic contains an internal contradiction. Before, when I was an atheist, I had no real basis for believing that, except for the same reason I didn’t believe in aliens. And yet, you and I both really, utterly, deeply believe that mathematics is elegant, gorgeous, and free from contradiction. If it’s just a handy dandy tool that makes us more likely to pass on our genes, then that is one huge coincidence. (Why should the conditions of our world be such, that having brains capable of perceiving flawless mathematics gives us a reproductive edge? We don’t have perfect vision or speed or digestion or anything else. Why is math so elegant?)
On the other hand, if the entire universe was created by an omniscient and rational Being, who also loved us and created us in His own image, then the existence of mathematics makes sense. I grant you, I can’t explain where the Being came from or His properties, but given my metaphysical view, the existence of consistent arithmetic pops out nicely. For the secular humanist, mathematics itself remains a puzzle to be explained.
Bob, this is really a shame. You claim to have been a “devout atheist” but you fall prey to the most amusing of fallacious blunders. As someone who should know better, this is a shame.
Do you not see how “On the other hand, if the entire universe was created by an omniscient and rational Being, who also loved us and created us in His own image, then the existence of mathematics makes sense.” is completely begging the question?
You ask the question about mathematics regarding why it is sublime or beautiful and perfect, and categorically throw out “coincidence” (like yeah, cmon, PFFT, like someone could have stolen Jesus’ body – get with the program!) in favor of a designer. Why? You make the implicit assumption that mathematics MIGHT JUST AS NOT be perfectly free from error, internal contradictions, etc. and the best reason to explain why it is not is due to a benevolent creator who wanted to give us tools to survive. Left completely unmentioned is the WIDELY POPULAR philosophical belief – whether Neoplatonic or formalist – that mathematics is necessarily logically sound because it explains relations in reality, which is itself free from contradiction and paradox. Mathematics is an analytic a priori discipline, tautological by necessity – not by chance or because my “creator” desired it so.
“I grant you, I can’t explain where the Being came from or His properties, but given my metaphysical view, the existence of consistent arithmetic pops out nicely.”
Granted, I can’t explain anything really important or salient about my hypothesis (ie, its properties) but, if you trust me for just a teensy second, then everything falls into place (kinda)! Really Bob? Saying that “God” created mathematics or reality or existence answers just as little as if I were to ask “How did that red car get in the parking lot?” and someone answered “GM made it.” I hope you’re willing to really sit down and think with us about the metaphysical and logical implications of a divinely inspired mathematics, because this is a pretty bad sentence.
“For the secular humanist, mathematics itself remains a puzzle to be explained.”
Not really. The finer points of advanced mathematical deductions (and applications of advanced mathematics like chaos theory) might be puzzling, but the metaphysical nature of mathematics has been explained pretty well for a long time. I think we’re all on the same page that Plato was wrong about ontological mathematical objects, and the more Kantian view is correct.
“that mathematics is necessarily logically sound because it explains relations in reality, which is itself free from contradiction and paradox.”
That’s an excellent argument for God’s existence, Mattheus! You are truly earning what we pay you to pose as an atheist and put forward these arguments. (Don’t worry about me blowing your cover — everyone will take this as a joke.)
First of all begging the question isn’t the most humorous fallacy by any means. Contradiction is way funnier.
Also your implication that either Bob should be shamed or is a liar is pretty insulting, and that’s right from the get go.
“On the other hand, if the entire universe was created by an omniscient and rational Being, who also loved us and created us in His own image, then the existence of mathematics makes sense.”
That’s not begging the question, if anything it’s a false dilemma followed by throwing out the other option without sufficient proof, or perhaps a strawman. But begging the question is just flat out wrong way to view it, whether he is right or wrong. Guess what fallacy you committed? STRAWMAN!
“Indeed, do molecules exist? Physicists will say that they aren’t really “solid” things either; they’re mostly empty space. If you really push it, according to cutting edge theories matter itself becomes more and more like an idea, rather than that “hard stuff” that’s “really” “out there” as opposed to the “not as real” stuff that’s in our minds.”
Hegelian idealism doesn’t fit traditional Christianity very well. If you want to argue that we are all essentially ideas – and maybe even further that God himself is an idea – then you’d have a heck of a time explaining how all these crazy theologians and logicians like Aquinas and Descartes were wrong about dualism.
“If you want to be really basic about it, the notion of a physical universe is a theory that we use to explain the more fundamental sensory data that we experience. After all, we might all be in The Matrix.”
Is reality or religion the Matrix? The joke runs both ways.
“Beyond the tremendous burden of walking around with that worldview, Karl’s statements are (again) arbitrary and unscientific. He says these things won’t be in a “spirit occupying another plane,” but modern science doesn’t tell us that. And to tell people that “in a relatively short period of time” entropy will engulf everything we care about is pretty close to demonstrably false. I mean, relative to what?! Karl is saying humanity will necessarily be extinguished, when every possible reference point is also extinguished. Short of eternity, what could be longer than the maximum age allowed by the laws of physics?”
His worldview seems to be physicalist and determinist. That’s not such a burden. In fact, it’s the only defensible worldview that can be supported by modern science. Modern science also doesn’t tell us that there are no flying invisible unicorns, or tea pots orbiting Uranus – but the clear absence of any such data would very quickly lead one to act as if those claims were false. If you would like to offer evidence that spirits occupy other planes, there are millions of scientists waiting to experiment with your evidence.
And he wasn’t talking about entropy when he mentioned “in a relatively short period of time.” He just meant that he will die, all his students will die, and after a few generations they will be forgotten (probably). A few lifetimes is a relatively short period of time, Bob. But just in case he was referring to the estimated 200 million years until the Sun engulfs everything, he’s still right. 200 million years is a fraction of all existence – something like 3%. That’s short.
Your unicorn and teapot examples fail for the same reason the flying spaghetti monster argument fails. All such arguments are incoherent.
http://centanium.com/2012/01/whos-afraid-of-flying-spaghetti-monster.html
What part of modern science supports physicalism?
“Your unicorn and teapot examples fail for the same reason the flying spaghetti monster argument fails. All such arguments are incoherent.”
“What the example shows, or is supposed to show, is that unless and until we have evidence that something is real, we should believe that it is not real.”
Actually what the flying spaghetti monster was intended to do was show the lack of any definitive criteria that grounded the Kansas State Board of Education’s decision to allow creationism to be taught alongside evolution. The creator of the FSM basically said “If creationism can be taught, then why not flying spaghetti monster as well?”
Side note: The creator of the FSM mail of course received death threats by some of the more “serious” creationists: http://www.venganza.org/category/hate-mail/
You interpreted the teapot argument of Guttenberg thusly:
1. We should believe that the Flying Spaghetti Monster does not exist.
2. Unless the non-existence of any given thing should always be presumed, we should not believe that the Flying Spaghetti Monster does not exist.
Therefore,
3. The non-existence of any given thing should always be presumed.
4. God is a given “thing”; therefore, the non-existence of God should be presumed.
This is all wrong. The argument is actually:
1. Modern science cannot directly prove a negative.
2. “Flying spaghetti monster does not exist” is a negative.
3. Modern science cannot tell us that there is no such thing as a flying spaghetti monster.
The additional argument he made:
“but the clear absence of any such data would very quickly lead one to act as if those claims were false. If you would like to offer evidence that spirits occupy other planes, there are millions of scientists waiting to experiment with your evidence.”
is an argument about how people act, not what they claim to know. He is saying that without any evidence, people will act as if the evidence doesn’t exist. That is not a claim to knowing that God does not exist. He is asking for the evidence that backs up creationists who DO claim to know that God exists.
There is nothing “incoherent” about that.
There is a corollary argument to all this, that Isaac Asimov began by asking “What happens when an irresistible force meets an immovable object?” He showed the inner contradiction of this question by referring to Einstein’s mass-energy equivalence, and showed that a universe with an immovable object cannot also contain an irresistible force, and a universe that contains an irresistible force cannot also contain an immovable object.
This argument relates to claim creationists often make when they say the universe contains entities that have the power to choose what they do (humans) and that the universe contains an omnipotent/omniscient deity that has the power to choose everything that occurs (God). If choice resides with humans, then that is an “immovable object” in relation to everything else that is not human. If choice resides with God, then that is an “irresistible force” in relation to everything else that is not God. Since immovable objects and irresistible forces cannot co-exist such that it makes sense to the human mind, then either humans have conceive of themselves having choice and God does not, or they must conceive of God having choice and humans do not. We cannot coherently regard both ourselves and things not ourselves having the power of choice in what we do.
Going even further, I share the view of compatibilists and other free will philosophers that we cannot even coherently regard ourselves as NOT having choice, and that we are past causally determined. We can’t pragmatically coherently make sense of acting towards something that we already consider to be past and settled. Deliberation, which is a vital and indispensable feature of action, cannot make pragmatic sense unless we presume our choices are not settled. In fact, to even deliberate whether would should accept determinism, would right away commit ourselves to rejecting it.
What type of compatibilism do you support?
“then you’d have a heck of a time explaining how all these crazy theologians and logicians like Aquinas and Descartes were wrong about dualism.”
Dualism is only one philosophical strain in theism. There has always been a strong current of philosophical idealism as well.
“Information is itself not a physical thing. For humans to perceive it with their sense organs, it must be instantiated somehow, I grant you. But the information itself is more than the physical components that represent it.”
But this doesn’t show that information is a “non-physical thing.” Physicalism doesn’t have to punt when it comes to theory of knowledge; there are perfectly good explanations of information in the non-reductive sphere (see Mary’s Room).
Sorry for my posts being all over the place. It’s 3 am and my mind is jumping. As a final word, I’m really curious about when you “used to be” an atheist. A lot of these arguments are very tired and I expect that as an atheist you would not have been satisfied with the answers you gave tonight. So either you have grown in intelligence beyond what you yourself would recognize (I doubt it – I mean, do I have to bring up liquidity preference again?) or you’ve fallen prey to fallacies you probably laughed at before.
As to Your last point (since to me the rest doesn’t have anything to do with god): whether human beings are capable of conceiving flawless mathematics or not is a completely different point than whether mathematics itself is flawless. If anything, there is ample evidence that most human beings across human history have been pretty bad at math. The only thing that is required to develop a half-decent understanding of math over time, however, is basic logic – and this, I think, does provide an evolutionary edge, both biologically and in society.
Also, coincidences do happen; if You choose a random real number between 0 and 1, the probability of choosing any one a priori is 0, but one is chosen in the end. Your line of reasoning sounds to me a bit like ‘an extraordinarily unlikely thing has happened, therefore it requires additional explanation’ – well, not necessarily, things that are extraordinarily unlikely happen all the time. Maybe evolutionarily developing a mind capable of reason and logic (or, alternatively, if we want to make the more realist argument, the spontaneous coming into existence of an orderly world) is one of them. And since in order to make the kind of arguments You are making, You have to be able to use some degree of reason and logic (or the world has to be orderly), the probability of the the rationality of the human mind (the orderliness of the world) on the condition that one is able to consider the question in the first place is actually very high – interpret that as You will, but to me the fact that only outliers would be able even to consider their outlierliness suggests that the ‘coincidence’ point of view (which I do not necessarily share) is completely unproblematic.
On the other hand, You’re right that the assumption that we won’t run into a contradiction in arithmetic is still somewhat arbitrary, however:
1) arbitrary assumptions about the orderliness of the world (or nature thereof) have been overturned before – in the XXth century we’ve learned that what most people thought about physic was false in a literal sense, so it’s not really an assumption that the secular humanist is necessarily committed to having
2) unless You have positive proof of the existence of god, You’re simply replacing one arbitrary assumption with two others: that such a god exists; and that the worldview assuming the existence of this god is elegant, gorgeous, and free from contradiction (this is still a prior assumption You have to make, otherwise any implication You make from the existence of this kind of god is suspect, because all implications are suspect). Without an additional argument, I don’t see how this is not pointless fluff.
“You’re simply replacing one arbitrary assumption with two others: that such a god exists; and that the worldview assuming the existence of this god is elegant, gorgeous, and free from contradiction”
There is nothing arbitrary about these “assumptions” — see, for instance, Anselm.
Arbitrary in the sense of not based on scientific grounding, rather than without any reasoning or thinking and just shooting from the hip.
It is arbitrary, from a scientific point of view, to postulate the existence of a deity through faith, and in Anselm’s case, it is arbitrary to postulate that some empirical concept in one’s mind necessarily exists in reality.
It is even more arbitrary to go from Anselm’s arguments to God being a Judeo-Christian God, and thus arbitrarily obey all the biblical nonsense of heaven and hell and killing people who work on the sabbath and brides who aren’t virgins.
It is the height of arbitrariness to justify one’s Christian faith by referring to past theologian’s arguments for God’s existence that don’t actually presuppose any one particular ultimate deity based religion over another.
Molecules do exist. If you have any doubt, just try to inhale carbon monoxide and see what happens to you. But I am told you cannot describe them without a huge amount of mathematics. Mathematics is all about entities which, apparently, have none of the properties ascribed to bits of matter (mass, charge, velocity, etc.). That’s why physics handbooks are full of maths. So reductionist physicalism (aka materialism) is a bit puzzling, at least to me. It looks like it’s self-defeating. Occam’s razor can be handled with too much confidence. It is certainly more economical to posit the existence of just one type of entities, material entities. But what if these material entities cannot be described without recourse to entities which, to the best of my knowledge, seem to be… immaterial? I have read a book by one of the good materialist philosophers, Mario Bunge (himself a physicist), in which he explains his position. A striking blind spot in his book was just that: he says that you can’t do physics without maths. Of course you can’t. What follows from that? The problem was not adressed at all.
“Trees are just another set of molecules”. Yes, and so are human beings, including you, Karl Smith, my eloquent acoustic pusher of molecules. Now why do lay preachers bother us with the Bill of Rights, the Universal Declaration of Human Rights and other silly sets of molecules? Remember, we’re just TSMs (Transitory Sets of Molecules) ourselves. Do TSMs have “rights”? Let’s be sensible, let’s be consistent. Hem, can TSMs be “sensible”? Terrible headache.
“Molecules do exist. If you have any doubt, just try to inhale carbon monoxide and see what happens to you.”
Wow, that is the worst proof of the existence of molecules I have ever encountered. Do you think non-atomists such as Aristotle and Descartes didn’t think poisons existed?!
Strictly speaking, you’re entirely right, of course. I was just trying to say that matter is not just “more and more like an idea”, as Robert Murphy put it, and my wording was hasty.
Mattheus, I have many problems with the things you have written here, but let’s focus on just one:
Left completely unmentioned is the WIDELY POPULAR philosophical belief – whether Neoplatonic or formalist – that mathematics is necessarily logically sound because it explains relations in reality, which is itself free from contradiction and paradox.
So Steve Landsburg can relax? He doesn’t need to worry that mathematics could conceivably contain an internal contradiction?
What about Marxism? The Marxists use their theoretical apparatus to explain relations in reality (means of production relates to legal structures, etc.). Since reality is itself free from contradiction, we can conclude that Marxism itself is necessarily free from internal contradiction. In fact, a Marxist shouldn’t be surprised by this result.
Bob,
The contradictions in Marxism are between the desired ends and the means of reaching them. There is no contradiction between the means of Marxism and the actual ends achieved. As Mises argued, Marxism is not compatible with reality.
Jonathan wrote:
The contradictions in Marxism are between the desired ends and the means of reaching them.
Uh oh, looks like someone hasn’t been reading enough Mises. First of all, what you are saying here would merely mean Marxism doesn’t “work,” in the same way that a rain dance doesn’t work. That’s not what anyone in any other conversation would mean by “Marxism contains internal contradictions,” certainly not Mises.
No, Marxism contains actual contradictions. For example, they talk about the progressive immiseration of the proletariat, and yet they also talk about workers being paid a subsistence wage. Contradiction. No value judgments involved, or recommending policies to achieve certain ends. Just in the Marxist’s positive description of reality, there is a contradiction right there (assuming a Marxist actually holds both propositions).
Bob,
You aren’t disproving me, your supporting my argument — and you don’t even realize it.
No kidding! That’s exactly what I said! The contradictions in Marxism are between the ends sought and the means employed. Marxism is not a path to the means sought by its proponents. This is a contradiction in the theory of Marxism, not in reality.
Right, there’s a difference between the Marxist description of reality and actual reality. But, that the Marxist description of reality is wrong doesn’t mean actual reality is contradictory, which is what you are trying to claim (erroneously).
By the way, your equivocation with the rain dance (that is, what you interpret my argument to be) is just as wrong as the rest of the comment.
Jonathan,
You wrote “That’s exactly what I said! The contradictions in Marxism are between the ends sought and the means employed”
I think you missed Bob’s point. He is saying that, according to Mises, Marxism contains internal (or actual) contradictions irregardless of its means-end framework and its conformance to ‘reality.’
Marx said “workers live on subsitence wages under capitalism” (p) AND “their that wages progressively fall under capitalism” (q). This is the actual contadiction; both (p AND q) can’t be true . You don’t have to look at the ends actually achieved by Marx’s ‘means’ to infer this contradiction.
Hence, Bob’s reference to rain dances. A rain dance is a ‘contradiction’ in the means-end sense; rain dances do not bring rain. But, saying “rain dances bring rain” is not an actual contradiction. Saying “rain dances bring rain” and that “rain dances don’t bring rain” is.
Richard, when you puts quotes around words and attribute them to somebody, those should be words they actually said or wrote! Marx initially thought wages would fall under capitalism, but abandoned that view as the facts came in.
Gene,
Apologies – my use of quotes was out of line.
“irregardless”
cringe
I would take a shower accept my hot water heater is broken and makes me loose money.
Your sense of humor at my expense is very much appreciated.
(I did laugh)
But my sense of humor was at Richard’s expense (who wrote “irregardless”). I was giving you a high-five, I thought…
“This is a contradiction in the theory of Marxism”
Which is a part of reality.
“they talk about the progressive immiseration of the proletariat”
Marx and Engels did NOT mean by this that the proletariat could not have their wages rise! They meant that, relative to the capitalists, the proletariat would become poorer and less powerful. There is no contradiction here. (They also thought the subsistence wage could change over time.)
“Marx and Engels did NOT mean by this that the proletariat could not have their wages rise! They meant that, relative to the capitalists, the proletariat would become poorer and less powerful.”
No, that is what Marx’s followers said Marx meant!
They only tried to advance that story after it was shown that real wages systematically grew in capitalism.
It isn’t what Marx himself actually said about what systematically happens to wages in capitalism. He never spoke of mere relative impoverishment, he made it quite clear he spoke of absolute impoverishment. It was supposed to be one of the reasons why the proletariat class would rise up and abolish capitalism. What proletariat would rise up and abolish capitalism because they had 2 yachts and 4 mansions, while the capitalists had 20 yachts and 40 mansions?
Marx harped on “subsistence wages” for a reason. He didn’t mean “what the poorest happen to be earning.” He meant SUBSISTENCE wages, or that which enables the worker to work in the first place.
Marx held that real wages tend to remain at minimum subsistence in capitalism, and he also said that real wages tend to fall in capitalism. He did contradict himself at two different points in his life.
“There is no contradiction between the means of Marxism and the actual ends achieved. As Mises argued, Marxism is not compatible with reality.”
If you’re talking about Marx’s metaphysics, then it’s not compatible with reality. It contradicts itself. Marx was a Historicist. Mises refuted Historicism.
Marx held that ideas are formed by technology and superstructure. As Mises also showed, and Rothbard later on, that contradicts the reality that technology and superstructure are a product of ideas, and don’t determine them.
Maybe you meant the political aspects of Marxism.
I meant the economic aspects. I.e. that a society where the means of production are collectively owned can efficiently allocate resources (or allocate them efficiently enough to achieve the desired utopia Marx had in mind).
OK, I guess it was hard to tell, since “not compatible with reality” sounds awfully close to a metaphysics argument. But then again, economics and metaphysics overlap when it comes to human action, so I guess it depends on which aspect of reality we’re talking about.
If mathematics are a gift from God, did it come in a bundle with logic?
I think math has at least one big flaw in it, and it is that “plus” is discriminated by a privileged “minus” in case of multiplications and divisions. Hence you have the problem that you cannot extract the root of negative numbers, which led mathematicians to invent their imaginary friend, the imaginary unit…
So unlike logic, math is far from perfect… Unfortunately I was not able yet to make a better more equal-rights based math that treats “plus” and “minus” equally and still allows you to calculate correctly without a need for crazy imaginary units.. 😉
imaginary numbers are as real as real numbers (and much more useful)
Show me -3i of anything real that is distinguished from 3 of that real something.
First I didn’t question that and secondly my post wasn’t really serious.
(Though this shouldn’t mean that I didn’t try once out of curiosity to find out what happens if you apply equal rules for minus and plus when multiplying… In short, it didn’t work..)
HAHA! Math Troll!! 🙂
First time I get the honor of being called a Troll.. Thanks Matt. 🙂
@ Bob,
Sorry for abusing your in fact quite serious blog post with nonsense like this..
Jonathan wrote: If mathematics are a gift from God, did it come in a bundle with logic?
Yes. Is that supposed to be a joke that embarrasses my position? They’re both gifts from God, as is reason, as is love, and as is my son.
Jonathan do you agree that Mattheus is spouting nonsense when it comes to his defense of the internal consistency of mathematics?
I don’t know what Mattheus’ argument is. But, why couldn’t logic and math simply be a product of the human mind, which is in turn the product of evolution?
I have asked mathematicians whether math is invented or discovered. They have all answered by way of Euler’s identity: exp(pi*i) – 1 = 0, saying that no one could invent that, therefore math is discovered.
Discovery means it is not of the human mind, but it is existing in nature for humans to explore.
Logic, by implication, is the same.
Another way to think of it is that math and logic are human (and therefore conceived in a way comprehensible to humans) of describing laws that exist in nature. And the laws are outside of humanity but humans are subject to those laws. And math and logic (and praxeology, yay!) are ways of describing those laws in human terms.
Jjoxman,
I can agree with that.
“And the laws are outside of humanity but humans are subject to those laws.”
More or less Bob’s argument, no?
Indeed.
“Discovery means it is not of the human mind, but it is existing in nature for humans to explore.”
One can “discover” what the mind has the ability to do and what it is about by virtue of being a mind. Discoveries don’t automatically mean it is not of the human mind. The human mind can “discover” its own talents and nature that it did not know before. Discovery is what ALL learning presupposes, of external reality AND of the mind itself.
“Logic, by implication, is the same.”
The same way of being of the human mind (and reality), yes.
“Another way to think of it is that math and logic are human (and therefore conceived in a way comprehensible to humans) of describing laws that exist in nature. And the laws are outside of humanity but humans are subject to those laws. And math and logic (and praxeology, yay!) are ways of describing those laws in human terms.”
If humans are subject to external laws, wouldn’t at least some of those laws be inside humans? The laws of thermodynamics are definitely inside and outside of humans. So is gravity (for some people, haha, actually no, it’s for everybody).
I think you are hung up on the physical differences of space. When I say “of the human mind” or “not of” I don’t mean it is physically outside humans or some such.
What discovery means is that the physical laws precede humans and are not created by humans. Computers are invented and created not discovered. But the physical laws that electricity obeys are discovered.
The math and logic constructs we create are invented, but the underlying laws being described are discovered. I hope that clears things up.
I think you are hung up on the physical differences of space. When I say “of the human mind” or “not of” I don’t mean it is physically outside humans or some such
Well then I recommend you consider looking at what you said again.
Would you agree that “not of the human mind” means that which is not of the mind?
Would you agree that identifying a one concept, makes it distinct? Would you agree that identifying a distinct concept implies an identification of two separate concepts, namely, the one in question, and everything else?
Would you (finally) agree that identifying two distinct and separate concepts immediately presupposes there is space separating them?
I hope you can see how what you said would lead someone to think spatially.
What discovery means is that the physical laws precede humans and are not created by humans.
What physical laws can possibly be created by humans?
Discovery, to me, just means something that is true first became known by a human.
Computers are invented and created not discovered. But the physical laws that electricity obeys are discovered.
No silly, God invented computers. It was his plan all along.
The math and logic constructs we create are invented, but the underlying laws being described are discovered. I hope that clears things up.
Yes, it clears up the fact that your position is that mathematics and logic are nothing but arbitrary intellectual conventions and rules that may or may not “describe” reality. I vehemently disagree with that position, but thanks for clearing things up.
Aren’t they fundamentally the same thing? i.e. Isn’t math ultimately about logical relations between statements?
So the statement 2+2=4 is true by definition; you’re basically saying that (1+1) + (1+1) = (1+1+1+1).
People who think math is about numbers are missing the point, it’d be like saying that economics is about money
That’s why I asked the question.
This is worse than anything Tom Hickey and co. have written. Where’s the old attorney when we need him?
Thanks MamMoTh. Most readers will take your empty mocking to be circumstantial evidence that I’m on to something.
Of course, that’s what’s really troubling.
I’ve got time for one more counterattack on Mattheus. Here goes:
[Karl Smith’s] worldview seems to be physicalist and determinist. That’s not such a burden. In fact, it’s the only defensible worldview that can be supported by modern science. Modern science also doesn’t tell us that there are no flying invisible unicorns, or tea pots orbiting Uranus – but the clear absence of any such data would very quickly lead one to act as if those claims were false. If you would like to offer evidence that spirits occupy other planes, there are millions of scientists waiting to experiment with your evidence.
OK Mattheus, so you’re saying modern scientists don’t believe in something, unless it is physical? They don’t believe in mathematics, they don’t believe in the conservation of energy, they don’t believe in “spooky action at a distance” as epitomized in the EPR paradox…?
No, that’s not what you mean. Obviously those non-physical things are fine. What you mean is, the vast majority of modern scientists don’t believe in a literal interpretation of the New Testament. Great. But let’s not mix up that much more modest claim, with the nonsense that they don’t believe in non-physical things having any impact on the material world.
And [Karl Smith] wasn’t talking about entropy when he mentioned “in a relatively short period of time.” He just meant that he will die, all his students will die, and after a few generations they will be forgotten (probably).
I was giving Karl the benefit of the doubt. You’re telling me, that when his students were wondering whether they should be optimistic or pessimistic, he was clinching the argument by informing them that they weren’t personally immortal? It takes knowledge of entropy and information theory to know that we are all going to die, probably within the next 100 years?
I’m assuming Karl was saying a bit more than what you are ascribing to him.
OK Mattheus, so you’re saying modern scientists don’t believe in something, unless it is physical? They don’t believe in mathematics, they don’t believe in the conservation of energy, they don’t believe in “spooky action at a distance” as epitomized in the EPR paradox…?
Mathematics and logic are the necessary superstructure of reality – they are categories of the synthetic a priori, as Kant would argue.
So right – empiricists cannot experimentally verify causality. Not because it’s non-physical, but because causality is itself necessary to make any experiments at all. Scientists presuppose mathematics and logic are true because they have no discernible method of disproving them (the idea of disproving logic sounds silly).
Conservation of energy and “action at a distance” are empirical laws. I can understand how scientists used to cast aspersions at people who believed in that – but the truth values of mathematics and logic cannot be denied, because denial entails a performative contradiction (unlike denying conservation of energy which may some day not hold).
What you mean is, the vast majority of modern scientists don’t believe in a literal interpretation of the New Testament. Great. But let’s not mix up that much more modest claim, with the nonsense that they don’t believe in non-physical things having any impact on the material world.
I’m saying the vast majority of moderns scientists, to the extent they are intellectually consistent, have every reason to disregard claims about non-physical entities. It would be absurd for someone to claim that they are demon possessed and, when asked for evidence, they respond “You can’t prove I’m not!!” The onus of proof is on the one who makes the positive claim. If atheist scientists came out and definitively said there is NO god – I would challenge them to provide reasoned argument and evidence for the absence of a being. Because they’re making a positive claim about reality. The solution is not to emphatically argue there is or there isn’t – because neither side can be defended empirically – but to be agnostic about these things.
One more reason to offer why I disagree on the existence of supernatural and non-physical entities is the problem of causal interaction. I simply cannot understand how something immaterial (God, Jesus, the divine) can physically interact with material objects, even in principle. How is it that Casper can catch a baseball and walk through walls?
I’m assuming Karl was saying a bit more than what you are ascribing to him.
Well, you know him better than I do.
“It would be absurd for someone to claim that they are demon possessed and, when asked for evidence, they respond “You can’t prove I’m not!!””
That’s exactly why you said scientists assume that mathematics and logic are true! Your reality certainly contains contradictions!
Weak
The difference between asserting mathematics is true and asserting you are demon possessed is mathematics and logic produce testable predictions.
IF logic is true and mathematics is true then X will happen. For instance, if Einstein was correct in his theory, then putting a clock in orbit will cause it to tick slower (by an exact specified amount) than a clock on the earth’s surface. This prediction has been confirmed and we use the fact every day in our GPS phones.
This fact (among countless others) was derived using mathematics and logic. So I’d say we have pretty good evidence that mathematics and logic are good models of reality. If someone doubts logic and mathematics are true then it should be very difficult for them to ever get on a plane, or even make a phone call.
Now let’s move on to the demon possessed person. IF I am demon possessed, then what ??? We have no testable prediction, just a bunch of people spouting about things they cannot prove or know to be true.
Adam Hickey wrote:
If someone doubts logic and mathematics are true then it should be very difficult for them to ever get on a plane, or even make a phone call.
So Steve Landsburg probably insists on taking the train, I mean walking, when he goes to conferences?
Really guys, you’re not arguing with the dumb Christian who thinks evolution is goofy, now you’re arguing with professional mathematicians who are also atheists.
“But the thing is, Bryan’s statement is actually silly, even if we don’t believe in the afterlife. Does Bryan, the tenured college professor, really think he needs to inform a former plantation slave that life isn’t fair? So my point is, whether you believe in the afterlife or not, Bryan’s statement makes little sense. It’s not that Bryan observed that there’s no afterlife, and so now he empirically can conclude, “Too bad, that guy’s hypothesis was falsified.”
I don’t think Caplan thinks he needs to inform the slave that life isn’t fair. I think he was saying the argument of the slave betrays the argument that life isn’t fair. Most people who make predictive arguments that imply a fair life are betraying the dictum “life isn’t fair.”
“My next blogger is Karl Smith, who is the James Joyce of economics blogging. In this post, Karl makes an analogy of the economy as a giant forest, and talks about why we should try to save the trees. He writes:
[W]e shouldn’t sit by while a new virus sweeps through and destroys the trees we love. In perhaps the grandest sense we could say, that yes these trees may die but we don’t worry because eventually they will be replaced by other trees in a never ending circle of life.”
“This is very true. But, we care for and love these trees — and that matters. On a deepest level it matters because our emotions are the ultimate source of value. At their core the trees are just another set of molecules. They are beautiful because they are beautiful to us.”
Rationalist philosophers, such as Rothbard, would have a field day with Karl Smith for saying “emotions are the ultimate source of value.” Rothbard (and Mises, Hoppe, etc) heavily criticized Hume and others for believing that reason is a slave to the passions.
If emotions are the ultimate source of value, then there is no argument that appeals to and is grounded in reason that can be made to refute anyone’s beliefs. That means Smith cannot say any other economist is wrong, according to Smith’s appeal to logic and economic science. He can only ever say they have different emotions than he does.
Can emotions be right and wrong? Based on what? He can’t say “emotions”, for that begs the question. If he says an emotion is right or wrong and bases that judgment on reason, then he would be nullifying his own dictum that emotions are the ultimate source of value.
“It’s the part I put in bold that concerns me. This is materialism in all its beauty / ugliness, depending on your value system.”
It isn’t a question of value, it is question of metaphysics, of ontology.
“First, note that this is a completely arbitrary statement on Karl’s part. Does the Pythagorean Theorem exist? It doesn’t consist of molecules.”
The Pythagorean theorem is a logical relation inherent in a concept which does have a counter-part consisting of, or relating to, real matter/energy. A real world right triangle would obey the theorem.
“Indeed, do molecules exist? Physicists will say that they aren’t really “solid” things either; they’re mostly empty space. If you really push it, according to cutting edge theories matter itself becomes more and more like an idea, rather than that “hard stuff” that’s “really” “out there” as opposed to the “not as real” stuff that’s in our minds.”
Mostly empty space means not all empty space.
So….you’re saying you would agree if someone said “God isn’t “really” “out there”, God is really “not as real” stuff in your mind”?
Consciousness implies being conscious of something. It is impossible for a consciousness to be conscious of nothing except itself.
“If you want to be really basic about it, the notion of a physical universe is a theory that we use to explain the more fundamental sensory data that we experience. After all, we might all be in The Matrix.”
That’s just pushing the objective grounds one step back, but without eliminating it. If this universe is “The Matrix”, then in order for that to make sense to us, it would have to be run by real entities, computer programmers, in an objective world of their own. It would be impossible to comprehend Matrix simulations of Matrix simulations of Matrix simulations of Matrix simulations of….etc, all the way down in an infinite regress. For if all simulations are simulated, what is a “simulation” at all? Simulations only make sense to us if we can distinguish and conceptually separate “the simulation” from “that which is being simulated”. If we lose the latter, we cannot comprehend the former.
“Yikes! Talk about a blogger needing God! Karl, please entertain the idea that there reputedly was a man who said many things that you would agree are very wise and good, and that this same man reputedly said paradise awaits us if we don’t reject it.”
So one who “needs” God is one who philosophically accepts temporal limitations on human life.
Yes, one “needs” God as a premise in order to believe the temporal limitation isn’t there and is just an illusion.
For Platonists, especially Plotinus, and St Augustine (who is the main source for the doctrines that most of today’s Christians worship, and whose writings were heavily influenced by the Platonists such as Plotinus), the contingency of human existence – or the “deficiency” of human existence for some people, as their philosophy is a rejection of the reality of humanity and thus a rejection of reality and thus a rejection of their own ability to reason – is most obvious in its temporal nature, not only in the fact that humans have a start and end point in their life, but also in his living through time at all.
This contingency, or “deficiency”, in man’s nature can be too intolerable and depressing for people to accept. The drive for removing this alleged defect and find a part of humanity that simply “is”, and “is” such and such independent of time and undifferentiated with respect to the rest of physical reality. This is the source of philosophy and of religion. To find an immutable, unchanging, eternal character to human existence that transcends temporal and spatial limitations that accompany a contingent existence.
Those who want to escape from such reality, who find human contingency as too intolerable and too depressing for them that it must be transcended, find refuge in accepting religious doctrines as true. The drive for freedom from all contingency is inherent in human life, and that is why religion is almost universal and atheism is in the minority (currently).
I expect religious belief to cease being popular once (if) humans become immortal. Only then will there be no more temporal limitation on human life, and thus no more temporal limitations in psychological need of transcending. As long as humans remain subjected to temporal limitations, this fact alone will continue to remain too intolerable for people to accept, and religious doctrines will act as a placeholder. It’s similar to how abused children and spouses “think of happy places” while they are being abused. The temporal and spatial limitations in human life makes many people think they are being punished for something they did wrong. Who or what would subject human life to such an unfair existence? We only live for 85 years or so, and then that’s it? How cruel! The concept of “original sin” anyone?
Or, we can be 100% materialists, know that nobody, nothing, is “punishing” us for being alive for only 85 years, know that human life is so very special that out of a universe of seemingly no life (that can be detected as of yet with current technology), there is this one small blue planet about 2/3 of the way out from the center of a galaxy, in a giant supercluster of galaxies, that contains life that can be conscious of itself. For materialists like me, knowing that physical matter is capable of being conscious of itself, such that I am able to type that on a computer, sending that information to other physical matter, provides me with feelings and thoughts that no religious text will EVER come close to doing. The wonder and awe that I feel and know being a full fledged materialist I could never get by reading 2000 year old ancient texts written by intellectually and emotionally impoverished people who could not tolerate the world they lived in and the reality of their contingency.
I was a Christian before, and I call tell you that I could never feel as joyous and free and in awe of the opportunity my life represents, until I ceased using those religious placeholders. I have come to conclude that religion is a cop out for those who need to believe their lives are not “wasted”. By accepting that one only has 85 years of consciousness, after which the physical matter and energy separates and the consciousness is lost, makes it very VERY tempting to deny that this is the only chance one gets and that human life that contains pain and suffering and mistakes and all the rest, are really just “tests” for some reward that will be given after. If a life containing pain was given, then surely a “just” reality would reward the people, right?
“Beyond the tremendous burden of walking around with that worldview”
Do you see what you just said? You said it’s a “burden” to have that worldview. In other words, for those of us who know that this life is the only time and place where our consciousness will exist, to you that seems like a “burden” that needs to be transcended. Why not accept it as reality, which cannot be anything other than what is, such that the “burden” is really just you yourself as you see yourself in the world, where negative feelings are derived by refusing to accept what is true?
I don’t consider it a “burden” at all. I consider it a tremendous “gift” inherent in the natural order. I say “gift” in quotes to find a way to relate the feeling to you since you’re a Christian, who views his life as being given to him by another conscious entity that you name “God”, but in reality I don’t even have a word for it. I think Buddhism comes closest with “Nirvana.” It is a state of thinking free from suffering or pain. Accepting one’s temporal and spatial limitation reality makes it impossible to think suffering or pain. Denying one’s temporal and spatial limitation makes it very easy to think human existence itself induces suffering and pain for no conceivable reason, and thus form the concept of a rewarding God for no conceivable reason.
“Karl’s statements are (again) arbitrary and unscientific. He says these things won’t be in a “spirit occupying another plane,” but modern science doesn’t tell us that.”
One can’t prove a negative. Science cannot even in principle directly tell us what doesn’t exist. It can only tell us what exists, and by way of indirect proof, tell us what doesn’t exist.
Modern science, indeed any science, has not proven that there exists a “spirit occupying another plane.” It can’t prove it one way or another anyway. It is not “unscientific” to reject what cannot in principle be established through logic-based or empirical-based evidence. It might be unreligious, but it’s not unscientific.
“And to tell people that “in a relatively short period of time” entropy will engulf everything we care about is pretty close to demonstrably false. I mean, relative to what?! Karl is saying humanity will necessarily be extinguished, when every possible reference point is also extinguished. Short of eternity, what could be longer than the maximum age allowed by the laws of physics?”
I think Smith meant relative to time periods relating to larger than human cosmic events, like galaxy formation (tens of billions of years), planet formation (billions of years), etc.
Hardly anything tangible is left of the Roman Empire, and that was just 2,000 years ago. We humans are getting better at creating things that last longer and longer, so that humans have decreased local entropy more and more at the expense of non-local increases in entropy, but on cosmic time scales, like billions of years, I don’t think any reasonable person would insist that whatever exists a billion years from now, will contain ANYTHING tangible that you and I and everyone else, cares about now. It will almost certainly all be gone. The only thing I can see remaining for that long are ideas (of course some ideas we currently have will almost surely be lost as well, and Christianity will almost certainly be relegated to the dustbin of world religions, just like many religions prior to the “big three” are forgotten, but still tabulated for information and museum purposes.
Immortal entities which evolved from humans living a billion years from now (humans probably won’t even exist at that time, as evolution by natural selection will have transformed humans into another species entirely) will probably look back at 2012 AD human Christians and see entities who wanted to be immortal but couldn’t be because of lack of technology and knowledge. So they will tabulate “Christianity” as “one of the earliest attempts at thinking about and desiring immortality in conscious entities that we evolved from.”
“Of course, the other problem is that Karl is here betraying the materialism from his other post. Information is itself not a physical thing. For humans to perceive it with their sense organs, it must be instantiated somehow, I grant you. But the information itself is more than the physical components that represent it. (Read Gene Callahan’s great post on these themes.)”
Information presupposes materialism. It is a phenomena of material objects. It is not something “beyond” materialism. It is not “more than” the physical components. It is OF the physical components. You relating an idea to someone else is fully explained by one physical entity bringing a certain physical orientation of vibrating physical air, or physical paper, or physical computers and ethernet cables, through one’s actions, such that the other entity is physically affected by it, such that their brains are altered and thus contain a brain pattern corresponding to the new orientation of physical matter brought about by the first person’s actions.
Ideas are patterns of physical objects (brains). Information is just another word for what is known and what can in principle be known through altered physical brain states that correspond to the patterns of physical matter and energy.
“Now we’re moving on to a really fun one. In this post, Scott Sumner was criticizing the Rothbardian view of the Great Depression. Sumner was arguing that the Fed couldn’t possibly have caused an inflationary boom in the 1920s. In the comments I asked him to clarify one of his arguments that amazed me, and he said:”
“I’ve shown there was no inflation as the term was defined at the time. I’ve shown that there was no alternative non-inflationary policy as understood by policymakers at the time, including those in the 1920s who claimed the Fed was too inflationary. It makes no sense to argue things were inflationary because M2 went up, if M2 didn’t exist. There are no policy implications. M2 was an idea invented much later.”
“One doesn’t have to be a Christian to see the non sequitur.”
Why is Sumner’s (terrible) arguments even in this post about bloggers needing God?
“You’re right, Steve: I don’t think arithmetic contains an internal contradiction. Before, when I was an atheist, I had no real basis for believing that, except for the same reason I didn’t believe in aliens. And yet, you and I both really, utterly, deeply believe that mathematics is elegant, gorgeous, and free from contradiction. If it’s just a handy dandy tool that makes us more likely to pass on our genes, then that is one huge coincidence.”
It’s not a “huge coincidence.” It follows exactly from the structure of our minds, which is a product of evolution by natural selection. It’s only a “coincidence” if you have already accepted human life to be a coincidental creation from an unknowable God entity.
“(Why should the conditions of our world be such, that having brains capable of perceiving flawless mathematics gives us a reproductive edge? We don’t have perfect vision or speed or digestion or anything else. Why is math so elegant?)”
As Kant showed, it’s because mathematics is a product of the way in which we MUST understand the world, due to the structure of our minds. People engaging in mathematics aren’t perfect, they are not infallible, they are not infinitely quick. Mathematics is based on repetitive operations of a praxeological entity.
You Christians are something else. When people are stupid and can’t explain the world around them, you say that what they can’t explain is due to God and hence God exists. When people are intelligent and can explain the world around them using their minds, e.g. mathematics, you say that the reason they can explain the world around them is due to God and hence God exists.
You’re asking all these questions as if you are being honest in not having an answer. They’re like rhetorical self-fulfilling prophecy declarations. I’m reminded of Bill O’Reilly: “Tides go in and tides go out. You can’t explain that”, then he goes ahead and explains it to himself by saying God did it.
“On the other hand, if the entire universe was created by an omniscient and rational Being, who also loved us and created us in His own image, then the existence of mathematics makes sense.”
“I grant you, I can’t explain where the Being came from or His properties, but given my metaphysical view, the existence of consistent arithmetic pops out nicely. For the secular humanist, mathematics itself remains a puzzle to be explained.”
How can you believe to be in an intellectually higher position by putting your inability to “explain” something on a concept that allows you to say “He did it and I don’t know how, but I know he did it, therefore my explanation is superior”. That’s totally fake. You’re not claiming anything other than “I don’t know why mathematics works.” You aren’t explaining it by referring to something else you can’t explain. Saying “God did it” is indistinguishable from “I don’t know.”
And mathematics is not a puzzle for secular humanists. We know that mathematics (in other words, logic) works because our minds are logically structured. Our minds are one of a thinking and hence acting entity. The gap between mind and reality is bridged when you understand thinking to be a property of certain material acting objects. The structure of such thinking is how reality behaves because our minds are a part of reality itself.
Of course, if you ask me why is reality so structured, I will say I can’t explain it, but given my metaphysical view, the existence of a relationship between mathematics and its success in human life pops out nicely. For the religious Christian, as well as all other humans, existence itself remains a puzzle to be explained.
Mises agrees with Hume that reason is merely instrumental to our desires. He did not believe, which Rothbard does, that reason can provide a basis for a rational system of ethics.
But I really like this post. I think the Kantian metaphysical and epistemological framework needs to be a lot more mainstream.
Mises agrees with Hume that reason is merely instrumental to our desires.
I guess if you interpret “subjectively desired goals” as “passion”, then yes, reason would be an instrument in Mises.
But Mises didn’t agree with Hume:
Mises wrote in HA:
“The rationalist philosophers themselves were always intent upon showing
the boundaries both of aprioristic theory and of empirical research. The first representative of British political economy, David Hume, the Utilitarians, and the American Pragmatists are certainly not guilty of having exaggerated the power of man to attain truth. It would be more justifiable to blame the philosophy of the last two hundred years for too much agnosticism and skepticism than for overconfidence in what could be achieved by the human mind.”
and later:
“We interpret animal behavior on the assumption that the animal yields to the impulse which prevails at the moment. As we observe that the animal feeds, cohabits, and attacks other animals or men, we speak of its instincts of nourishment, of reproduction, and of aggression. We assume that such instincts are innate and peremptorily ask for satisfaction.”
“But is different with man. Man is not a being who cannot help yielding to the impulse that most urgently asks for satisfaction. Man is a being capable of subduing his instincts, emotions, and impulses; he can rationalize his behavior. He renounces the satisfaction of a burning impulse in order to satisfy other desires. He is not a puppet of his appetites. A man does not ravish every female that stirs his senses; he does not devour every piece of food that entices him; he does not knock down every fellow he would like to kill. He arranges his wishes and desires into a scale, he chooses; in short, he acts. What distinguishes man from beasts is precisely that he adjusts his behavior deliberatively. Man is the being that has inhibitions, that can master his impulses and desires, that has the power to suppress instinctive desires and impulses.”
Hume of course would have vociferously disagreed with Mises here.
I think it’s the case that, like you said, Mises, unlike Rothbard, did not accept reason can establish a rational ethics. But that is a far cry from holding reason to be a slave to the passions. What if one’s “passion” was to establish a rational ethics? Mises the utilitarian would have said sure, reason can be used as a means to achieve that goal. He unmercifully attacked Hegel and other seeming objective ethicists for being arrogant, but it was because they stepped outside the praxeological playground and believed themselves to be aware of what a universe-God wanted.
“Hume of course would have vociferously disagreed with Mises here.”
Um… no?
Um…why?
“as their philosophy is a rejection of the reality of humanity and thus a rejection of reality and thus a rejection of their own ability to reason”
Major, you’re the champ of saying stupid things, but this one tops them all.
Coming from you, that means I in the hemisphere of truth and logic, rather than….whatever it is you call your crazy madhouse of imbecility.
Completely unrelated question. I’ve just never read the Bible, so have no idea either way. Does Jesus ever consider himself the Messiah?
This is not an attempt to criticize Christianity, by the way. It has more to do with Judaism than anything else.
Jesus knows he’s the son of God and sent by God as the savior of mankind.
In Judaism, as far as I know, anybody who claims to be the Messiah is a false Messiah. Is this an applicable criticism of Jesus?
I think Jewish people think so.
To get an idea of their mindset at the time, how do modern day people react to people who call themselves the Messiah? Not too well, right?
I’m not sure, but more than a reaction, I think that’s somewhere either in the written scriptures or in the oral traditions.
Maybe so. My knowledge of Judaism is what I picked up from being a Catholic. So, nil.
The Mosaic writings, Psalms, histories, and prophecies all predict a coming savior (Messiah in Hebrew, Christ in Greek). The Jewish scholars in Jesus’ day took passages like “The Lord your God is One,” while not minding passages like “Let us create man in _our_ image.” They rejected the idea of a God who could and would exist in three persons yet still remain One. Jesus was crucified for claiming to be God (“I and the Father are One,” “I am,” and other phrases which caused the Jewish religious rulers to want to stone him, throw him off a cliff, or crucify him – see Matthew 26:62-66). He pointed to his miracles as proof that he had been sent by God. The prophets had done this. He took it a step further and claimed to be able to forgive sins. He said, “Before Abraham was, I am.” I got a little tangential, but the point is not that they rejected the prophecy of a Messiah, but they did not understand a) that the Messiah would be God and b) that the Messiah must die as a propitiation for sin.
Jonathan, yes. More than that, Jesus claims He is God. Here He says that He is one with the Father, and here He says He was alive before Abraham and refers to Himself in the same way that the LORD of the Old Testament does. (It was because of this blasphemy that the Jews tried to stone Jesus after He said this. They knew full well what He was getting at.)
Let me try to clarify some things here.
(1) Steve Landsburg, as well as a bunch of the greatest mathematicians in human history, worry about whether arithmetic and higher bodies of mathematics contain contradictions. They sure hope not, but they realize you can’t prove such a thing. For sure, you can’t use a given axiomatic system to prove that that particular axiomatic system doesn’t contain a contradiction–that would be assuming the very thing in question.
(2) I argued that Landburg’s own metaphysical views give him little reason to believe in the consistency of arithmetic. If math just “seems right” to our minds, because our brains evolved in a certain way giving us a reproductive advantage, then we might very well find a contradiction in arithmetic next Thursday. Just like, our eyes help us survive, and yet people can experience optical illusions. The way our eyes/vision work is good enough.
(3) Mattheus said that my argument is nonsense. Since reality has no contradictions, and since mathematics is a system of thought describing reality, therefore mathematics obviously won’t contain contradictions.
(4) I pointed out that this was clearly a bad argument. Of course the human mind can invent systems of thought, describing reality, that contain internal contradictions. For example, Mattheus presumably agrees with Mises that Marxism is contradictory.
(5) Jonathan said that Marxism was only contradictory in terms of its recommended methods for achieving certain ends.
(6) I pointed out that no, that’s not at all what Mises meant when he said Marxism was contradictory. If it were, then we would also have to say people doing a rain dance were engaged in contradiction, or that (a new example) doctors who used leeches in the 19th century had medical knowledge that was internally contradictory. This isn’t at all what Mises meant. He meant, for example, that in one spot the Marxists assert a law of history that the proletariat gets progressively poorer, while in other places the Marxists say that the worker is paid the bare minimum necessary to survive. (Whether Gene Callahan is right in saying actual Marxists didn’t do this, is irrelevant. The point is, *that* example given by Mises is an internal contradiction. That’s a totally different thing from saying, “Socializing the means of production won’t lead to a higher standard of living.”)
In summary, it certainly is possible to have a system of thought–even an a priori one–that is internally contradictory. We don’t know if arithmetic is contradictory; that’s why Steve Landsburg (who teaches this stuff at the graduate level) “hopes to God!” that it doesn’t.
I see your point now (thank you Richard Moss). I was looking at a different contradiction, which may not be a contradiction at all.
i.e. much paraphrased and generalized:
Marxists say (A) communal ownership over the means of production will lead to (i)higher wages, (ii)a world post-scarcity, or (iii) whatever they actually said.
Reality is (B) communal ownership will lead to capital consumption, thus falling real wages and a world of deepening poverty.
I wasn’t looking at internal contradictions, rather contradictions with reality. And, when I summoned Mises, I meant his argument of resource allocation (economic calculation).
Steve Landsburg, as well as a bunch of the greatest mathematicians in human history, worry about whether arithmetic and higher bodies of mathematics contain contradictions. They sure hope not, but they realize you can’t prove such a thing. For sure, you can’t use a given axiomatic system to prove that that particular axiomatic system doesn’t contain a contradiction–that would be assuming the very thing in question.
You can’t even in principle claim a contradiction exists without presupposing the very axiomatic system you say contains a contradiction!
I argued that Landburg’s own metaphysical views give him little reason to believe in the consistency of arithmetic. If math just “seems right” to our minds, because our brains evolved in a certain way giving us a reproductive advantage, then we might very well find a contradiction in arithmetic next Thursday. Just like, our eyes help us survive, and yet people can experience optical illusions. The way our eyes/vision work is good enough.
The only way that anyone can even claim to be able to know of a contradiction, would be through the very logic you say is only hypothetically true. This is a priori true. Logic is therefore not hypothetically true. It must be treated as categorically true.
We can’t even in principle comprehend a different logical structure to our minds such that one is resorted to calling the logic being addressed as only hypothetically true. Logic has to be considered apodictically non-hypothetically true if any refutations or identifications of contradictions are to be taken as superior to the original logic that allegedly contains the contradictions such that it ought to be rejected in favor of whatever it is you are proposing.
The analogy to optical illusions is flawed, but fitting.
It is flawed because the meaning of optical illusions only makes sense if there is some understanding of what one is actually observing in itself, that explains the observation being an optical illusion as opposed to not being an optical illusion, which of course presupposes that you are cognizant of what reality actually exists that elicits the illusion.
It is fitting because it aptly shows the contradiction in postulating that logic can be contradicted, when the very meaning of “contradiction” presupposes the very logic you claim is only hypothetically true and not apodictically true.
If you say logic is only hypothetically true, then you must treat any “contradiction” of that logic as only being hypothetically true as well, but then you’d then have to explain why anyone should adopt that versus the other allegedly hypothetically true system, which of course means you would have to explain the difference between hypothetically true statements and non-hypothetically true statements such that hypothetically true statements even make sense, and not only that, but should be adopted instead of non-hypothetically true statements. After you do that, you’d of course get dinged for using non-hypothetically true logic, thus revealing another pragmatic contradiction.
Logic simply cannot be considered only non-hypothetically true and potentially containing flaws or contradictions. It leads to incoherence and pragmatic contradictions.
You interpreted Mattheus’ response to be “since mathematics is a system of thought describing reality, therefore mathematics obviously won’t contain contradictions.” You said this argument is a bad argument, on the basis that “of course humans can invent systems of thought”, and these systems are possible to have no reference in reality and are contradictory, thus it is wrong to say that just because we think it, it has to be apodictically true.
That response interprets Mattheus, via Kant, to be advocating a form of idealism. Such a response is expected, because Kant’s epistemology does seem to be idealist. But it really isn’t. Kant showed hints here and there that he grasped the praxeological, and hence non-idealistic foundation of mathematics and logic, but he unfortunately didn’t go the extra steps to make it explicit. Mises did. In the world of academia, Mises’ contributions to philosophy are very under-appreciated, even by Austrian economists!
In summary, it certainly is possible to have a system of thought–even an a priori one–that is internally contradictory. We don’t know if arithmetic is contradictory; that’s why Steve Landsburg (who teaches this stuff at the graduate level) “hopes to God!” that it doesn’t.
We can know that mathematics and logic are non-contradictory and apodictically true, because of the very fact that any attempt to refute it, will require it to be true. Any and all arguments about anything whatever, cannot violate logic or else it wouldn’t make any sense and thus cannot even be categorized as a valid refutation. Valid refutations depend on logic. If you ask that others question logic, then there is no reason to even accept your argument that logic is hypothetically true instead of non-hypothetically true.
MF wrote: We can know that mathematics and logic are non-contradictory and apodictically true, because of the very fact that any attempt to refute it, will require it to be true.
MF, arithmetic can have an internal contradiction. It is an open question. Arithmetic and logic are not the same thing. You discredit the performative contradiction literature when you so carelessly apply it in areas where you apparently don’t know what you are talking about.
(Really, guys: Are you saying Landsburg and the great mathematicians whose work he is summarizing, are running around in circles? They are worried about something that a few moments’ reflection shows can’t possibly hurt them?)
“MF, arithmetic can have an internal contradiction. It is an open question. Arithmetic and logic are not the same thing. You discredit the performative contradiction literature when you so carelessly apply it in areas where you apparently don’t know what you are talking about.”
Wow, what a hasty accusation. I am fully aware of Hilbert’s second problem. It’s not “careless” to apply self-referential logic to mathematics. Mathematics is entirely BUILT on first order logic, which is itself grounded in action. We can use what Newman called “meta-mathematical logic”, or what I would call praxeological self-reflective logic, to prove that mathematics is internally consistent. We don’t have to stay within the mathematical axioms we are trying to show are consistent.
Godel showed that the Peano axioms cannot by themselves prove that they are internally consistent. But that doesn’t mean that the question of whether mathematics is internally consistent is “an open question.” It is an open question for mathematicians yes, but there is no reason for us to limit ourselves to mathematical axioms and symbology. They are but one language in the system of logic.
When I kept saying “mathematics and logic”, “mathematics and logic”, I did not intend for that to mean ONLY mathematics on the one hand, and ONLY logic on the other hand, separately. I meant logic and all of its sub-branches, including mathematics.
Who says we have to remain in the world of mathematical axioms? The whole point of my insistence that mathematics and logic are a priori true is not that I claim to have refuted Godel, it’s that mathematics is but one system of logic, and if Peano mathematics is going to be considered inconsistent, such an argument will have to rely on the very logic that I claim is a priori true. No mathematician can ever deny this without contradiction.
In all systems of logic, my position is that they are all ultimately grounded in action. Action cannot be mathematically represented by Peano axioms. Expecting to be able to prove that Peano mathematics is consistent by divorcing it from self-reflective action is stripping away the very foundation of logic and thus making it impossible to prove it is consistent.
What Hilbert asked, and what Godel and Gentzen partially answered, has to do with whether or not mathematics is by itself internally consistent. That is not something I intended to say is the case or not the case, nor is it something that I even require to know that mathematics and logic are a priori true and contradiction-free.
“(Really, guys: Are you saying Landsburg and the great mathematicians whose work he is summarizing, are running around in circles? They are worried about something that a few moments’ reflection shows can’t possibly hurt them?)”
The world’s great mathematicians treat mathematics as merely a verbal game of symbols and semantic rules. I don’t know of any great mathematician who accepts that mathematics is grounded in action and thus can never stay in the idealistic world only, but must be connected to non-mathematical action.
This quest you are on of attacking the epistemological foundations of those who are attacking Christianity, in order to cast doubt on their arguments, in the hopes of therefore bringing them down the level of doubt and ignorance required to believe in God, is very apparent, and I am fully aware that this is going to be something you NEED to attack relentlessly, until you are satisfied you have the upper hand in casting doubt on everyone and everything, so as to make your faith not seem so outlandishly misguided.
Okay great, now I understand where and why you disagree with me.
Let me ask you – is it possible that praxeology proper, as opposed to those who practice it, can be wrong or contain contradictions? Can logic itself – and not certain imperfect logicians – make mistakes?
Of course they can, which is why I don’t get too bent out of shape when LK criticizes Mises on praxeology. He was an imperfect practitioner of an a priori discipline – but the discipline itself is a priori.
Likewise, I can argue that mathematics is and always will be internally consistent because it is defined to always be internally consistent, even though certain mathematicians may err in the application of the axioms.
“Of course they can” = “of course the practitioners can be wrong while the discipline is sound”
I REALLY need to proofread.
This seems to me to be the Walrus and the Carpenter (Satan and Christ), the result being that the oysters get eaten. It’s mostly arbitrary definitions. There are no two apples that are identical, so how can there be two apples? There is this apple and that apple, if I accept that apples are a class, but the idea that they represent two apples is a matter of faith. I suspect that there are no two electrons that are identical. Numbers are useful for certain predictions, but they do not exist in “reality” except as we assign them. Similarly, the question, “Does God exist?” is meaningful only if we define the term. I don’t believe in any of the definitions of God I have ever heard of, but I do know that a Spirit exists, since I identify Spirit as Experience and I am experiencing right now. I don’t have to know whether I am physical or in the Matrix to know that I am experiencing this right now. This seems to me to negate any idea that God does not exist. I exist, since I am experiencing; if there is no other God, then I must be God, by default.
MF, Mattheus, and others: Skim this Wikipedia entry. Hilbert’s challenge to mathematicians went like this (paraphrasing): Sure, we all use arithmetic and it clearly seems to “work.” Nobody has ever stumbled upon a contradiction. But what if we tried to represent all of our (sometimes intuitive) knowledge of arithmetic in a formal, axiomatic, deductive system? Could we then be sure that it would be impossible to “prove” two statements in that system, where these statements contradicted each other?
This is a very deep question and if you say, “Duh, of course you couldn’t do such a thing, because the very attempt to do so would prove that the task is impossible,” then you have missed the problem.
To repeat my point on this issue: Steve Landsburg and, I suspect, 95% of all professional mathematicians, would be horrified if it turned out that arithmetic contained some obscure contradiction that didn’t affect what we use “math” for in everyday life. Landsburg might literally become suicidal.
And yet, given his metaphysical views in general, I see no reason for him to have such faith in the consistency and elegance of mathematics.
MF, Mattheus, and others: Skim this Wikipedia entry. Hilbert’s challenge to mathematicians went like this (paraphrasing): Sure, we all use arithmetic and it clearly seems to “work.” Nobody has ever stumbled upon a contradiction. But what if we tried to represent all of our (sometimes intuitive) knowledge of arithmetic in a formal, axiomatic, deductive system? Could we then be sure that it would be impossible to “prove” two statements in that system, where these statements contradicted each other?”
Praxeologists don’t need to be limited to mathematical axioms. Of course if you strip away the axioms, and to a larger degree, logic, away from their grounding of action, then you will never be able to have a complete system that is consistent. Mathematics is rooted in our understand of repetition, which action presupposes. One can never establish that mathematics is consistent by stripping mathematics away from action, and treat them as freely floating propositions and arbitrary semantic rules. That is what today’s mathematicians do, and that is why Hilbert posed the question and why there is no universal agreement and why you read wikipedia entries and believe consistency in mathematics remains an open question.
It was never my position that mathematical axioms and semantic rules stripped from their grounding in action can be proven internally consistent. Of COURSE they can’t be proven consistent if one treated mathematics as an arbitrary system of rules and definitions that “seem to work really well.”
To repeat my point on this issue: Steve Landsburg and, I suspect, 95% of all professional mathematicians, would be horrified if it turned out that arithmetic contained some obscure contradiction that didn’t affect what we use “math” for in everyday life. Landsburg might literally become suicidal.
They will never find an internal contradiction in mathematics as long as mathematics is understood as but one system of logic which is grounded in action. What Landsburg is saying is that he hopes that mathematical axioms stripped from their grounding in action, and treated as arbitrary freely floating rules and symbols, will never be shown to be inconsistent. Yes, one should be mindful if mathematics is treated that way. It’s like stuffing gas into a tank, ignore the pressure gauge, and hope that the pressure inside isn’t enough to blow up the tank. His fear is derived from the fact that modern mathematicians do treat axioms as merely verbal stipulations and freely floating abstractions.
This is, incidentally, why some mathematics in the world is not useful for human life, and why other mathematics are useful for human life. The more abstract and arbitrary the system of rules, the further mathematicians go into a world of pure ideology, essentially religion.
And yet, given his metaphysical views in general, I see no reason for him to have such faith in the consistency and elegance of mathematics.
And the penny drops. I see what you are doing. You are trying to make it seem like FAITH underlies one’s convictions in mathematics and logic. That way, your own faith doesn’t seem so far fetched. Gosh can you be any more transparent?
This attack on human epistemology and your constant attempts to pigeon-hole all human epistemology into one of utter groundlessness, of pure faith, is so obvious. You just want to convince yourself, and others, using poor Landsburg and the entire field of mathematics, that humans cannot have an objective, God-free consistent and contradiction free knowledge about reality. Humans MUST be ignorant and MUST guide all their actions and base all their knowledge on faith. Only God is allowed to have objective consistent, contradiction free knowledge. Don’t want to offend the big guy, now do we?
Not only that, but you don’t seem to realize that by attacking the objective character of logic and mathematics, and basing it on faith, you are undercutting every single argument you have ever made against your intellectual opponents. For you have clearly used logic and mathematics every time you claim to have refuted them. But if all mathematics is based on faith, and thus if logic is based on faith, then how can you even claim to have refuted anyone?
You say I took a big dump on the performative contradiction literature when I “carelessly applied it” to mathematics. And yet for some “careless” reason, I am able to see the performative contradiction in your own arguments that you clearly are not able to see.
This is why I dislike the Sunday posts. It turns an economics blog that satisfyingly demolishes economic ignoramuses, into an idiotic festering stinkpile of contradictions and crap logic.
Maybe every Sunday I should just ignore the post completely and pretend they’re not there. I don’t see any way of settling differences of opinion when the common system of logic that all agreement, disagreement, and argumentation presupposes, is rejected, in favor of some silly faith based initiative.
MF wrote: I don’t see any way of settling differences of opinion when the common system of logic that all agreement, disagreement, and argumentation presupposes, is rejected, in favor of some silly faith based initiative.
For the record, you are the one who is changing the rules. You are going to redefine what “math” is so that your position holds.
When I was saying, “It is an open question whether math contains a contradiction or not,” I meant “math” as defined by mathematicians, not “an enterprise that is useful for human life and is grounded in the action axiom.”
I’m not even saying your approach is wrong, just that you are being like Gene Callahan when he says, “Everybody is a theist, because I define ‘God’ as believing the universe follows rules,” or whatever. That’s a valid position, but he can’t then go pouting, “Nobody will stick to the commonsense definition I’m using! What the heck!”
MF wrote:
They will never find an internal contradiction in mathematics as long as mathematics is understood as but one system of logic which is grounded in action….This is, incidentally, why some mathematics in the world is not useful for human life, and why other mathematics are useful for human life.
Ah, OK I see your position now. (I’m not even being sarcastic, though a bit condescending I grant you.) If mathematicians discovered an outright contradiction next Thursday, and 30% of them jumped off a bridge because they couldn’t stand to live in such a world, you would shrug and say, “Well that type of ‘math’ obviously isn’t useful for human life. It’s not real math.”
Have I correctly understood your position, MF?
Ah, OK I see your position now. (I’m not even being sarcastic, though a bit condescending I grant you.)
I said it before and I’ll say it again: I realize and accept that my approach is caustic, and very susceptible to being mocked, dismissed, etc, but in the long run, I have found it works for bringing out that extra bit of oomph that allows me (and almost always others) to tap into that intellectual spirit that is so often absent in sterile classrooms. If it makes me look like a fool, so be it. If it makes others look like fools, so be it. My goal is truth, not popularity, feeling good, or being respected.
If mathematicians discovered an outright contradiction next Thursday, and 30% of them jumped off a bridge because they couldn’t stand to live in such a world, you would shrug and say, “Well that type of ‘math’ obviously isn’t useful for human life. It’s not real math.”
I would say that you should reflect on what it is you are saying whenever you make any argument, then make sure you are cognizant of the logic you are presupposing.
Here you provide a thought experiment where mathematicians discover a contradiction in the set of axioms they originally took for granted. What Landsberg “hopes to God doesn’t happen.” I would say you ought to ask yourself how such a contradiction can even be comprehended and established by anyone. In other words, what are the grounds for knowing the contradiction even exists?
Then I will ask you to realize that THOSE grounds are what underlie all logic in general, and mathematics in particular if the concept of “contradiction” is even going to be let in through the door of the mathematics building. To even say that a contradiction exists in mathematics is already to presuppose that mathematics is logical and that the error was on the side of humans making a mistake, not mathematics which is logic codified and ultimately grounded in the reality of action.
The only other option, for humans, is to believe that mathematics is beyond logical criticism, and is somehow “true in itself” because it is grounded in arbitrarily stipulated axioms that are not ultimately grounded in action.
It would be like a mathematician saying to me “Oh, you say I am wrong to say 2+2=5? That if mathematics is grounded in counting, which is grounded in action, it is false? Well you silly naive fool, you should listen to Landsberg. In the mathematics field we stipulate any axioms we want, and according to such and such arbitrary stipulations, it is true that 2+2=5.”
This kind of mathematics is what I was referring to when I said the world’s great mathematicians treat mathematics as a game of manipulating freely floating concepts according to arbitrary rules. It is this kind of mathematics that I consider to be useless (in the pragmatic sense and ontological sense, not in the utility and pleasure sense).
I am trying to think of a good analogy to drive this home…hmmm…
OK, let’s consider economics why don’t we. Austrians hold that economics is grounded ultimately in action. Well, that doesn’t mean that everyone who “does economics” realizes the ultimate foundation of economics and grounds their arguments, i.e. constrains them, to action. Some economists like mathematical economists and econometricians, they simply postulate arbitrary assumptions, and using those freely floating abstractions, they manipulate them according to mathematical rules (which may or may not be grounded in action), and out pops their conclusion that the government has to spend $364,237,283 to avoid recession. This is useless economics because the concepts are not grounded in action. Some economists on the periphery, like the Austrians, who are not considered “great” by most economists, hold that useful economics is economics grounded in action, not freely floating conceptions that are manipulated according to conventional rules.
Major Freedom, one of my favorite parts of Human Action is where Mises tackles the polylogist dismissal of economics. He says (paraphrasing), “OK, so the polylogists dismiss the findings of Smith, Ricardo, and Menger by saying that these men are using ‘bourgeois’ logic and not proletarian logic. But that’s not enough to make their case. The polylogists must go further, and show what alternate axioms or rules of deductive inference apply in proletarian logic, and then show why–in that system–we can deduce that protectionism, minimum wage laws, expropriation of the capitalists, etc. all act to increase the standard of living of the workers.”
So I would like you to do something similar. Take a branch of mathematics that you agree might be internally contradictory because of the fragile way in which Landsburg or Godel would erect it, and then you build it instead on a praxeological foundation, and then prove that this alternate construction is incapable of contradiction.
If what I’m asking would take you 3 weeks to actually do, then at least sketch a demonstration of what you have in mind. I’m curious to see, for example, if you allow the complex plane to be counted as “real math,” things like that.
So I would like you to do something similar. Take a branch of mathematics that you agree might be internally contradictory because of the fragile way in which Landsburg or Godel would erect it, and then you build it instead on a praxeological foundation, and then prove that this alternate construction is incapable of contradiction.
Lorenzen and Wittgenstein is where I would begin. They both provide a Kantian interpretation of mathematics. Wittgenstein in particular wrote a “notorious paragraph” concerning Godel’s incompleteness theorem, that led me to asking this question:
“If Godel’s incompleteness theorem is true, would it be the case that if one were to use it as one axiom in a larger mathematical theory, that this larger theory would necessarily be incomplete or inconsistent as Godel’s incomplete theorem suggests is the case for all mathematical theories?”
If the answer is yes, then wouldn’t Godel’s incompleteness theorem be itself incomplete or inconsistent? If the answer is no, then wouldn’t Godel’s incompleteness theorem be contradictory?
I tried to take Godel’s theorem and then ground it in action, by USING the theorem in some larger argument (theory), rather than keeping it in some no man’s land of freely floating propositions as mathematicians love to do, in order to see whether the act of using it in a theory, doesn’t lead to a contradiction in what it claims to be the case for all theories.
I have asked many mathematicians, students, PhDs, you name it, and none knew of an answer. I don’t know the answer but I suspect that Godel’s incompleteness theorem really just shows us that a mathematical theory that is not grounded in action but is purely arbitrary axioms and rules based, cannot prove itself true. Well that seems intuitive, doesn’t it? All proofs require an actor who can prove things. Without an actor, proofs cannot occur.
All mathematical proofs therefore must be ultimately grounded in action, and not remain isolated in the floating concept world of arbitrary symbols and rules, where mathematicians typically remain, and where they hope that their theories will prove themselves without any grounding in action. This is what I consider to be a mystification of mathematics. It is why so many mathematicians these days are laboring away in their windowless offices doodling on paper, rather than applying them to real world applications. It’s also why, I think, so many mathematical economists are also laboring away in their windowless offices instead of making millions in the market. The last time they were unleashed, they mystified the housing market and turned prudent planning into an incomprehensible formulaic game that even the owners didn’t fully understand. They just hired the PhD math wizards and let them screw everything up.
If what I’m asking would take you 3 weeks to actually do, then at least sketch a demonstration of what you have in mind. I’m curious to see, for example, if you allow the complex plane to be counted as “real math,” things like that.
My thoughts on complex numbers would be that they are incredibly useful in practice, and are “closer” to a praxeological foundation than, say, infinite-dimensional manifolds which still remain in the classroom, and probably always will.
The complex plane is modified Cartesian coordinate system. It is geometric, and geometry is, you guessed it, presupposed for an entity acting in a world and so is grounded in action. The rules are somewhat different, but the operations in the complex system are very similar to real systems, which is why, I suspect, they have use in engineering, computer science, and other technical practical fields.
As for a rough sketch on how to ground complex planes on a praxeological foundation, I probably would first look up Wittgenstein and Frege and their criticisms of psychologism (not to be confused with polylogism). Identifying praxeological, rationalist foundations for things typically arise from a critique of mutually exclusive and competing doctrines.
Then I would consider the axioms and rules and meaning of the symbols, and see if any of them contradict the constraints imposed by action. I have already done this with the concept of infinity. The concept of infinity is beyond the constraints imposed by action, and so can never be known by humans and is ontologically meaningless. Every time infinity is used in mathematics, it is a placeholder for that which cannot be integrated into action, and must always be represented by finite counting estimates when explained and argued. It’s like saying “Imagine a point on a line moving towards zero, it gets closer and closer and closer (which is praxeology talking) to zero, and after passing through an infinite number of theoretical points (which is where we go beyond praxeology and have to “summarize” things in concepts such as infinity), the point gets to zero.”
They will never find an internal contradiction in mathematics as long as mathematics is understood as but one system of logic which is grounded in action.
Priceless! I’ve never seen such a ridiculous statement before in my life. You should publish a paper!
How is it ridiculous?
BTW general announcement: I am being somewhat pointed on this only because I think you guys are being a bit sloppy with details that we all agree on. I.e. I am throwing around terms like “nonsense” when I should be calmer, but to be clear I am not saying if you disagree with my overall position, you are speaking nonsense. I’m talking about much smaller issues, like whether it even makes sense to ask if arithmetic is consistent, if Marxism is contradictory, etc.
Very interesting, Dr. Murphy. May I suggest a different direction of approach?
The Egyptians defined Pi as 3.1, others defined it as 3, and Pythagoras spent years trying to find even numbers by which to measure right triangles.
Yet the ratio of the circumference to the diameter of a circle did not change, Pythagoras did not invent triangles with sides of 3-4-5 and 5-12-13. Nor did Einstein invent relativity, clocks change their rates under acceleration with or without his insights.
Facts exist independent of any proof, or even awareness, by people. Schrodinger’s cat is either dead or alive, we just don’t know which until the box is opened. It’s surprising the heat I get for saying that!
Mathematics is a science because it is tested. When a function is found incorrect, when reality does not match the predictions, it’s not reality that is questioned, it is the prediction. Someone made an assumption that was incorrect.
Austrian economics is based upon this same rigorous self-discipline. If reality does not comply with the theory, it’s the theory that is wrong.
Keynesians cannot understand “stagflation” because it doesn’t fit their theory, so they deny “stagflation” exists. Stimulus must work, so obviously none of them were ever done “early” enough or were “big” enough, no matter how many times stimulus fails. Etc.
I have the self confidence to admit that I just don’t know why things are this way or that way. I cannot explain capillary action, but water is drawn into small tubes whether I can explain it or not.
Certainly “God did it” is much, much easier than other explanations for phenomena, it removes the discipline of questioning the theory.
It also salves the tender ego by providing a catch-all, so that there is never a reason to say “I don’t know.”
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Have you ever heard about “Male Answer Syndrome”? It’s a survival trait, where a male will make up some plausible sounding theory on the spot on a subject he knows nothing about, in order to impress the Ladies and have a greater opportunity of procreation. This leads me to wonder, after so many generations of “Male Answer Syndrome”, if there is a genetic predisposition in the basic human make-up against ever admitting “I don’t know.”
“Indeed, do molecules exist? Physicists will say that they aren’t really “solid” things either; they’re mostly empty space. If you really push it, according to cutting edge theories matter itself becomes more and more like an idea, rather than that “hard stuff” that’s “really” “out there” as opposed to the “not as real” stuff that’s in our minds.”
And then of course there’s the truly baffling realization that empty space itself contains matter. Existence and nothingness are one and the same!!
Who knew economists could get so deep?
Bob, I like your blog, but … do I get it right that what your are basically saying is “we are not sure why mathematics is consistent, therefore God exists”? If this is the case, well, it is exactly the same as saying “we don’t understand X (substitute anything to your liking), therefor God exists”: I can understand the urge to prove the existence of God among some of the more evangelical types, but I am not sure if it can be done by similar “logical” statements.
David Kyjovsky:
Bob, I like your blog, but … do I get it right that what your are basically saying is “we are not sure why mathematics is consistent, therefore God exists”?
Nope, that’s not at all what my argument is. I am merely mentioning (a) atheists who deeply believe that mathematics is internally consistent have no sound reason for such a belief, and (b) it so happens that belief in a rational God solves that particular problem.
The fact that it works really well and hasn’t messed up yet implies that math is internally consistent. Just like the fact that no one has ever seen you eating a human being implies that you are not a cannibal.
Tzadik is that a joke?
As impoverished as the reasoning is, the “after X iterations of everyday observation, our conception of the world has demonstrated to be proven again and again, therefore our conception of the world is true” argument seems to me to be more or less the starting point for many modern layman atheists, especially when coupled with a belief in parsimony.
This is more or less Newton’s strong empiricism, without the benefit of Newton’s understanding that some problems remain out of the scope of empirical study.
Of course, we could have made a similar argument during the long ages where the geocentric model of the universe held sway, except that there were known holes in the theory. The apparent tightening up of the models of reality (even if in fact they remain at an insurmountable distance from a true understanding) we use pose a problem to the imagination.
No. What’s wrong with induction to figure stuff out?
What about grue and bleen?
it so happens that belief in a rational God solves that particular problem.
No, it doesn’t.
Exactly. It is an “answer” that is no answer at all.
Bob:
I’m very sorry that I don’t have time to respond in detail to your post and that I haven’t had time to read all of your comments. But let me say a few random things that might help people avoid some intellectual traps.
1) In any line of inquiry, one must start from somewhere. To me, and I think to most mathematicians, the natural starting point is the existence and basic properties of the natural numbers. There is a sense, then, in which my (and most mathematicians’) belief in the natural numbers is an act of faith, like your belief in God. I am prepared to argue at some future time that my act of faith is more defensible than yours, but I’m sure you’ll have worthy counterarguments.
2) A more-or-less equivalent act of faith would be the consistency of Peano arithmetic. But the existence of the natural numbers is not the same thing as the consistency of Peano arithmetic.
3) Once you believe in the existence of the natural numbers, I believe you can *derive* the existence of physical reality, so this requires no further acts of faith. (Here I am expressing a view that is not unique to me, but I have no idea how many mathematicians would agree.) That’s another thing my act of faith has in common with yours.
4) One wants to be careful about phrases like “the consistency of mathematics”. There are many inconsistent mathematical theories. What’s interesting is whether *particular* theories (such as Peano arithmetic) are consistent.
5) It is incorrect to say that there are no proofs of the consistency of Peano arithmetic. There is, for example, Gentzen’s proof. There is also the proof that says “The natural numbers exist; the axioms of Peano arithmetic are true when interpreted as statements about the natural numbers; because they are true, all of their implications are true; threfore they cannot imply a contradiction.” Each of these proofs, like any other proof of anything in any subject, of course relies on some prior belief (e.g. “The natural numbers exist”.)
6) Commenters who have suggested that the truths of arithmetic follow logically from standard axioms and defintions are badly mistaken. Godel’s theorem says precisely that this is not true.
7) Regarding the last few paragraphs of your post, I believe (for reasons I’ve expounded on at length elsewhere) that a) mathematics exists by necessity (much as, I suppose, you believe that God exists by necessity), that b) there is a very concrete sense in which the Universe is *made* of mathematics (which explains why it is so well described by mathematics) and that therefore mathematics is not a puzzle to be explained.
8) My brief scan of the comments failed to turn up the one that said something like “obviously mathematics is consistent; otherwise airplanes would crash” (something you quoted in a comment on my own blog), but if someone did say this, s/he is badly mistaken at multiple levels. First, it makes no sense to ask whether “mathematics” is consistent; it makes sense only to ask if a particular mathematical theory (e.g. Peano arithmetic) is consistent. Second, even if Peano arithmetic turns out to be inconsistent, that would not invalidate the existence of the natural numbers; it would just mean that we had been mistaken about some of their properties. Third, it’s entirely possible for an inconsistent theory to be perfectly useful in practice. The evidence for the consistency of mathematical theories comes from within mathematics itself, not from any applications.
9) On the other hand, I’m perfectly happy saying “obviously the natural numbers exist, otherwise there would be no reason for anything *else* to exist”. But here I am expressing my own view, not some universally accepted view among mathematicians.
10) But while I’m at it: I am utterly baffled by people who are willing to believe that there is such a thing as an airplane, or a rock, or a molecule, and not willing to believe that there is such a thing as the system of natural numbers. It seems to me far far more likely that my mind is fooling me about physical stuff than about arithmetic. (Though I don’t think either is very likely.)
11) In other words, if you’re pointing to things *outside mathematics* to justify your belief in mathematics, then I think you’re assumptions are shakier than your conclusions, which kind of renders your whole argument pointless.
Thanks for the very useful thoughts, Steve. I think you are basically saying, “You guys are all ignorant.” But, I at least am content that I was right, when I said they were speaking nonsense…
1) In any line of inquiry, one must start from somewhere. To me, and I think to most mathematicians, the natural starting point is the existence and basic properties of the natural numbers.
This is the philosophical starting point of existents. The natural numbers are such a clear cut contender for a starting point because at their most fundamental level, conscious entities receive raw data, complex raw data, of a myriad of separate entities. The only way to hold multiple entities at once is to find some commonality or commensurability between them. The commensurability between heterogeneous entities is that they are all unitary existents. They are all unitary, single entities. The commonality between a red apple and a red dress is that they are both red. Red, red. The most fundamental commonality between them is that there both single existents. The commonality is 1, 1. The single concept is “2 existents.” The commonality between a green apple, a red dress, a steak, a car, is that they are also existents. The commonality is 1, 1, 1, 1. The single concept is “4 existents.”
This is what everything has in common, and so begins how the consciousness integrates the universe’s heterogeneous concepts. Identifying entities is therefore a process of counting. e.g. “This first entity I observe is therefore an entity that is not me.” That leads to one and two. “This second entity I observe is not the first entity and it is also not me.” That leads to one and two and three. And so on.
The entire universe is a single concept that adds up every single existent. Just consider what you typically think of when you consider the entire universe. You systematically, but very quickly, roughly add everything that exists in terms of existents. You consider all of them at once, and the only thing that everything has in common that can enable you to do this, is that they are all single entities that in mathematics and logic we associate with natural numbers and counting.
The natural numbers are how an acting entity integrates heterogeneous concepts that come to the senses. A commonality among all of them is discovered such that they can all be grouped together in some way no matter what is being considered. The only way to group heterogeneous concepts into a single picture is by first grouping them in a particular way, namely, by realizing they are all singular existents. This is why we can add apples and oranges. We can say there are 1, 2, 3 apples here, and 1, 2, 3 oranges there, and there are 1, 2, 3, 4, 5, 6 apples and oranges. We can add heterogeneous existents! This is the origin of natural numbers and why acting entities perceive such usefulness in natural numbers. It is probably the most fundamental structure of ANY acting entity’s consciousness. It’s why babies LOVE it when you count before you make funny faces or feed them, and why they don’t like it so much when you just abruptly do things without prompting.
We count because we act through time. Action is the objective basis for sequential counting and thus the objective basis for the natural number system.
2) A more-or-less equivalent act of faith would be the consistency of Peano arithmetic. But the existence of the natural numbers is not the same thing as the consistency of Peano arithmetic.
Yes, “faith” is required if you refuse to ground the validity of natural numbers on the basis of action. If you just arbitrarily start with natural numbers without grounding them on an objective foundation like action, then sure, the only other option is “faith.”
3) Once you believe in the existence of the natural numbers, I believe you can *derive* the existence of physical reality, so this requires no further acts of faith. (Here I am expressing a view that is not unique to me, but I have no idea how many mathematicians would agree.) That’s another thing my act of faith has in common with yours.
This is just idealism that doesn’t address the foundation for why natural numbers and not something else. Physical reality isn’t “derived” from the use of mental tools like the natural number system. Physical reality DERIVES the natural numbers for acting entities. The entity perceives other things and hence can reflect on itself, which right away gives us the counting 1 and 1 existents, or 2 existents. Within the external world, there are a myriad of entities, which gives us the counting 1 and 1 and 1 and…sum total of existents the actor is aware of.
4) One wants to be careful about phrases like “the consistency of mathematics”. There are many inconsistent mathematical theories. What’s interesting is whether *particular* theories (such as Peano arithmetic) are consistent.
Inconsistent mathematical theories are theories that have been identified as contradicting the constraints imposed by the nature of action.
5) It is incorrect to say that there are no proofs of the consistency of Peano arithmetic. There is, for example, Gentzen’s proof. There is also the proof that says “The natural numbers exist; the axioms of Peano arithmetic are true when interpreted as statements about the natural numbers; because they are true, all of their implications are true; threfore they cannot imply a contradiction.” Each of these proofs, like any other proof of anything in any subject, of course relies on some prior belief (e.g. “The natural numbers exist”.)
Gentzen’s proof is not a full proof of the consistency of Peano axioms. It is a proof of a weaker form of them, and there are criticisms of Gentzen’s arguments. Followers and antagonizers of Gentzen will go back and forth until they ground their axioms in action.
6) Commenters who have suggested that the truths of arithmetic follow logically from standard axioms and defintions are badly mistaken. Godel’s theorem says precisely that this is not true.
Agreed. But I will argue that what Godel proved is that arithmetic cannot prove itself devoid of an actor that transcends the mathematical axioms. Proofs of all propositions, for any field of inquiry, require the existence of a transcendence beyond the rules themselves, and must include the actor making the proof. Even mathematical proofs require this. If action is not included, if the axioms remain freely floating and abstracted away from action, then they cannot prove themselves.
7) Regarding the last few paragraphs of your post, I believe (for reasons I’ve expounded on at length elsewhere) that a) mathematics exists by necessity (much as, I suppose, you believe that God exists by necessity), that b) there is a very concrete sense in which the Universe is *made* of mathematics (which explains why it is so well described by mathematics) and that therefore mathematics is not a puzzle to be explained.
To the conscious actor, yes, the universe external to itself is made of mathematics. But the actor cannot coherently regard itself as being entirely constrained to mathematical logic.
And I would be careful if I were you in saying that “the universe” is mathematical. Remember, you’re in it too. If you’re going to accept that Godel proved that the truths of mathematics do not logically follow from the axioms themselves, then you can’t say that it is true that “the universe” is entirely mathematical. For then you would only be contradicting what Godel proved.
You MUST take into account that you are an actor that transcends mathematical logic, but not praxeological logic of course, and that you can only say that everything other than the actor is mathematical. There needs to be something there to prove it all of course! Don’t step outside yourself and mistake yourself for being God and having a bird’s eye view of the universe.
8) My brief scan of the comments failed to turn up the one that said something like “obviously mathematics is consistent; otherwise airplanes would crash” (something you quoted in a comment on my own blog), but if someone did say this, s/he is badly mistaken at multiple levels. First, it makes no sense to ask whether “mathematics” is consistent; it makes sense only to ask if a particular mathematical theory (e.g. Peano arithmetic) is consistent. Second, even if Peano arithmetic turns out to be inconsistent, that would not invalidate the existence of the natural numbers; it would just mean that we had been mistaken about some of their properties. Third, it’s entirely possible for an inconsistent theory to be perfectly useful in practice. The evidence for the consistency of mathematical theories comes from within mathematics itself, not from any applications.
That second point you made tells me there is at least some hope for you. But I will go one step further. I will say that because counting arithmetic (natural numbers) is grounded in action, it will never be refuted. I claim to have this as objective knowledge and nobody will ever refute it. The only way to approach this is to either accept it, or be wrong.
9) On the other hand, I’m perfectly happy saying “obviously the natural numbers exist, otherwise there would be no reason for anything *else* to exist”. But here I am expressing my own view, not some universally accepted view among mathematicians.
This is Godthinking and is not justified for a non-God actor. You are saying there is no reason for why things are another way, as if things are what they are for a reason, as opposed to being absolutely primordial and without reason. It’s hard for people to grasp this because as actors, we cannot help but approach the world through purposeful behavior. To conceive of events that take place for no reason is why it is so tempting to ascribe to the universe a reason for its existence, i.e. God.
10) But while I’m at it: I am utterly baffled by people who are willing to believe that there is such a thing as an airplane, or a rock, or a molecule, and not willing to believe that there is such a thing as the system of natural numbers. It seems to me far far more likely that my mind is fooling me about physical stuff than about arithmetic. (Though I don’t think either is very likely.)
I think what they mean is that we don’t observe the numbers 1, 2, 3, 4 as physical objects or entities (well, other than semantic numbers drawn by people of course). There are 1, 2, 3 apples there, but we don’t “see” the number 3. We ascribe the symbol of 3 to the apples that do exist.
11) In other words, if you’re pointing to things *outside mathematics* to justify your belief in mathematics, then I think you’re assumptions are shakier than your conclusions, which kind of renders your whole argument pointless.
This is where you go so very wrong. Your first point above in 1) is that the natural numbers must be taken on faith. Faith, the last time I checked, is in fact “outside mathematics”. Now you’re saying it’s “shakey” to justify one’s belief in mathematics by pointing to something that is “outside mathematics”? How can you say it’s shakey, when that is exactly what you are doing when you use faith to justify your belief in mathematics?
It’s not “shakey” at all to ground natural numbers in praxeology, which is “outside mathematics.” You do act when you count, yes? It would be a performative contradiction to deny that you are acting when you are counting, yes? No acting entity that counts could ever claim that their action of refuting this praxeological foundation, is consistent, yes?
You’re wrong. We not only can, but we normatively should, point to foundations outside the scope of mathematical language to justify mathematics. Justification and proof are themselves actions! Mathematical symbols don’t prove themselves. Mathematical proofs REQUIRE action, which is itself not constrained to mathematical logic, but the larger, grander, praxeological logic.
Wittgenstein, not Russell, not Godel, not Hilbert, not Euclid, not Brouwer, not Plato, not Aristotle, not Frege, not Lewis, not etc.
Hilbert attempted to recast mathematics as nonsense standing on stilts, Godel showed that is must be nonsense. Wittgenstein explained how and why folks were positing the nonsensical stilts — and why they needn’t do so.
Most mathematicians are still stick in the fly bottle imagining they are walking around on Platonic stilts or Hilbert stilts or such.
They don’t get Wittgenstein — and don’t need to to get tenure and publish.
PS: One commenter writes: “The world’s great mathematicians treat mathematics as merely a verbal game of symbols and semantic rules. ” Clearly, this commenter knows nothing about how the world’s great mathematicians treat mathematics.
Steve, please, the man’s a Major. Show some respect.
“PS: One commenter writes: “The world’s great mathematicians treat mathematics as merely a verbal game of symbols and semantic rules.” Clearly, this commenter knows nothing about how the world’s great mathematicians treat mathematics.”
That commenter was me.
Clearly it is apparent that you also have no clue about the presumed epistemological foundation of the world’s great mathematicians. None of the “great” ones consciously ground their mathematics in action. Most are taught “the rules of the game”, not the philosophical foundations.
It’s well known that if we treat math as a bunch of axioms and inference rules, we can’t prove it to be consistent. What utterly baffles me is when people start to say, “Oh, yes, but if we treat math as something that humans invent or intuit or act upon, then everything’s all right” – as if people never produced contradictions.
Now, Bob says “if we treat math as something God invented, then everything’s all right”. True enough, as long as God exists and never produces contradictions. Now that, at least, is a reasonable idea. But for someone to reject God, treat math as a purely human invention, then express faith in its consistency? Bizarre.
Remember, then, Russel’s paradox – math was expressed through a set of axioms and intuitions, and then bang! A great big contradiction appeared in the middle of it. Nonetheless, balloons didn’t fall out of the sky, not did steam trains blow their engines. There’s no reason to think that our current understanding of the best way to talk about mathematics can’t suffer the same fate. I talk more about this at http://www.dr-mikes-math-games-for-kids.com/blog/2011/10/the-hitch-hikers-guide-to-mathematics/
Mike H writes: It’s well known that if we treat math as a bunch of axioms and inference rules, we can’t prove it to be consistent.. But this, under most reasonable interpretations, is not so.
First of all, there is no plausible way in which you can treat all of math as a bunch of axioms and inference rules. You can treat some *part* of math that way. And then whether or not you can prove it to be consistent depends on what part of math you’re talking about. There’s no problem, for example, with proving the consistency of euclidean geometry.
Steve Landsburg wrote:
First of all, there is no plausible way in which you can treat all of math as a bunch of axioms and inference rules. You can treat some *part* of math that way. And then whether or not you can prove it to be consistent depends on what part of math you’re talking about. There’s no problem, for example, with proving the consistency of euclidean geometry.
Steve, I realize this is akin to you challenging my statements regarding bald Irishmen, but I thought you were being a little unfair to Mike H in your latest comment here.
When the layman asks, “Do mathematicians know if math is internally consistent?” I think he means, “Everything that mathematicians do in their careers…Can they be assured that it is more rigorous than, say, what the physicists know about the laws of Nature?”
This, after all, was the promise of doing stuff axiomatically, right? That the mathematicians would have a much stronger degree of certainty in their field, than just about any other humans.
Now my admittedly fuzzy understanding was that sure, you could prove the consistency of things like Euclidean geometry, in the sense that if someone endorsed the axioms (which seemed pretty non-controversial), then it was easy to show non-contradiction.
But once you introduce things like existential and universal quantifiers (for others: the ability to say “there exists an X” or “for all X such that”) and couple that with the real numbers, then things start getting trickier. If you want to be able to embody all possible propositions in that system, then to prove consistency you have to rely on axioms which are not more obvious than the original statements themselves.
And I thought the work of Godel specifically said that sure, you can always make a meta-system that encompasses all of the propositions you have in mind, and prove the consistency of that sub-system, but by doing so you now have made it possible to state true propositions in the meta-language that cannot be proven to be true.
Bob:
1) Axiomatic systems have very little to do with the actual practice of mathematics. They are *models* of mathematical reasoning; they are not mathematical reasoning itself. I write down economic models all the time in which people care only about consumption and leisure; the act of writing down and studying such models does not in any way constitute a promise about how real people will behave.
In fairness, I suppose that people like Hilbert did have in mind something like the promise you refer to, but that was a very long time ago and I don’t think much of anybody thinks that way anymore.
2) Your reference to the real numbers is off base; you meant to say the natural numbers. I do not believe Godel’s theorem applies to the real numbers.
I’ll presume this was a typo. 🙂
3) The above quibbles aside, the gist of what you say is certainly correct.
PS: Why do smiley faces keep popping up in the middle of my posts?
You’re a happy guy?
(I think it’s because you are putting an “8” and then a closed parenthesis.)
It is a quirk in the commenting format. If you attempt to do an emoticon at the end of a sentence, it immediately gets pushed to the left on that line. I’ve learned to skip to the next line to use emoticons, and then skip another line before starting a new sentence.
PS – note that Steve Landsburg seems to treat humans as a purely mathematical invention, not the other way round.
I enjoyed the great comments on this question, as always; Bob has a way of finding the topics that I really care about and posting about them.
I’ve been spending some time with Edward Feser’s writings on teleology and asking myself a question.
Nihilism is the obvious, and I think probably unavoidable, worldview of strict atheism; i.e. there are only relative bases for morals, no “story from on high” to grasp. Atheists might say that their goal is survival and the protection of knowledge, because self-aware systems (like people, dogs, and maybe lobsters) work to ensure least the medium-term survival of information (i.e. genes, other things we care about). Of course, the long term worldview for atheism is not very cheery either, even if you factor in the possibility of quantum level effects appearing at the macro level being used to stave off entropy.
The question is this: What does nihilism actually offer as a worldview? Prudence, for example, is a virtue, and anybody can use it as additional or even basic (if they are a virtue ethicist) support for arguing a position. You don’t need to be an atheist or a theist to argue from prudence; in fact it stands as the core of an entire brand of ethics. I think that nihilists can actually adopt arguments like virtue ethics or even teleology, but their bases will now be relative, not absolute.
As an answer to the question, I received the response that “maybe it prevents us from making errors.” But I don’t think that nihilism does this. Empiricism and reason operate just as fluently for a theist as they do for a strong atheist. The key to whether this will prove ultimately well-founded or misguided rests on the assumption that we can construct a test of some of the classic metaphysical questions, such as “can we find all the basic foundations of a belief system, instead of just asserting their presence with reason,” i.e., proving to a skeptic that God does or does not exist, or proving all the basic axioms of a logically consistent system of arithmetic.
There is a reason that many people say that some questions are simply beyond the reach of science, and even if we look beyond that limit with a nihilist viewpoint, and find the nihilist viewpoint to be accurate, it offers no guidance on correct action. While this ultimately cannot bother the nihilist, it makes it much harder for the nihilist to argue from anything but an essentially ad-hoc position.
I think the break in views between the nihilists and other ethical thinkers is that nihilists seem themselves as “free” to rationally or humanely construct a system of ethics with some justification in whatever arguments seem expedient; others tend to be scared by this view as it affords no viewpoint, argument, or set of facts deference to set a basis for ethics.
The issue I can’t get around is that nihilism seems to assert something which isn’t testable, which seems contrary to the basic atheist methodology. Maybe it is merely a defect that theists don’t recognize this as a problem; maybe it is a wash.
Perhaps mathematics and reason offer a hand – we can (and should) have some conception of various types of infinity to talk about possible theories of the origin of the universe, for example; the conclusions remain intelligible, even if we reject them.
I think that you are wrong. I for one am an atheist, I would say “strict” but not sure what you mean by that. I might even say that I am an agnostic, being open minded, but it is a bit of alibistic attitude given that there is not a single sound argument for the existence of God that would be consistent enough to convince me. Anyway: I am sure, that most atheists are not nihilists in a sense that they have a pretty good set of moral and ethical values. Where those came from? Upbringing, education, thinking about the world, culture in a broad sense, and most are, I will hazard a guess, even natural to humans and sourced by evolution. In any case, most people come to the same set: do not do to others what you would not like being done to yourself, do not cheat, steal, do no unecessary harm. Etc etc. Religious people use the argument of the “source of ethics” as an argument for the existence of God and some kind of their moral superiority. I believe that it is pure arrogance, nothing more.
David Kyjovsky wrote:
[T]here is not a single sound argument for the existence of God that would be consistent enough to convince me.
David, just to be clear, do you mean to say “that I have yet heard” or do you mean the statement as it now stands?
Bob, obviously I cannot be sure that I have heard/read all of them. But: if there was one that strong, I suspect that it would be all over the Internet and other media, hotly debated by all kinds of people. Anyway, I consider myself open minded and not (too) arrogant, so I am of course ready to change my miind if I find an argument too persuasive and strong to ignore (it would be even pleasant, in a way, because the idea I am sure can alleviate existential anxiety, certain fears etc). My problem (one of them in fact) with the God hypothesis is this: it is dogmatic and it cannot be falsified (so it is not helping understanding of anything much). It can possibly answer all questions, therefore it is really not answering any. Last thing, pardon me if I am not clear enough, not being a native English speaker.
David Kyjovsky,
I was listening to a history lecture from The Great Courses, “Great Minds of the Eastern Tradition”, which I highly recommend.
By compressing such an expanse of time, basically all of written history east of the Black Sea, the lecturer inadvertently demonstrates how religions were simply made up on the spot over, and over, and over. Yet there were always people who would believe that this one, and none other, was the “truth”.
And since anyone who believes in this or that god generally does not believe in all the other gods that people have made up to avoid saying “I don’t know”, to be an “atheist” ends up being someone who just believes in one less god.
The religious drone on about how someone not subject to the will of a god has no way to measure their morality. Yet the greatest danger of religion is that it provides an “out” of the Non Aggression Principle. After all, it’s not aggression if one is doing God’s Will.
I think we’re reaching diminishing marginal returns on this one. Good stuff though, everybody.
MF let me leave perhaps a final thought: Believe me, I understand the appeal of the performative contradiction approach. I agree that it’s an incredibly strong argument to deploy against someone who denies the action axiom. Yes, someone who tries to disprove the action axiom is acting, and in that sense is engaged in contradiction just as surely as Barack Obama saying “no US president is capable of speech.”
However, if I show you some subtle contradiction within arithmetic, that doesn’t violate the action axiom. Depending on the contradiction, it might not even violate anybody’s intuitive notions of common sense, at least not directly, in the sense of “A and not-A can’t be simultaneously true.”
Since it seems you understand a lot about higher mathematics, MF, let me ask: Are you familiar with the Axiom of Choice, and the freaky stuff that it implies? That’s the kind of stuff I mean. You can pick some seemingly innocuous axioms in a formal, deductive system, and generate crazy implications that you didn’t expect.
So just because our attempt to formalize even simple arithmetic would seem straightforward, it’s conceivable that the axioms actually allow for a contradiction. If this were to happen, it wouldn’t render action itself nonsensical.
And finally: I don’t see what reason the atheist has, for thinking such a world is impossible. Yes, math “seems impregnable” to your mind right now, just like it really seems like there is water in the desert to your eyes when you see an oasis. Why would you think evolution gave you a brain that could discern objective truths about reality, as opposed to approximations that are good enough to outwit a lion?
“Why would you think evolution gave you a brain that could discern objective truths about reality, as opposed to approximations that are good enough to outwit a lion?”
Because rape is abnormal, and women like poetry.
“Evolution” is not a force, it is a result. There is no such “thing” as Evolution, there is only the innumerable natural selections made over time.
How many children survive, now, into adulthood to breed? Most, and only because human beings, deliberately or not, “bred for brains”.
Exactly. And also to be able to cheat other members of his social group, to be able to identify cheating, to be able to keep track of favours and counter-favours, simply: to build inside his mind a model of mabe dozens other minds and keep running this model, on and on. Compared to this task, modeling the physical environment and looking out for lions is piece of cake.
David, as much as I hate to say it, one word used to encompass what you describe is “politics”.
Yes, and it makes a lot of sense… succesfull politicians tend to have more sex, and that is all you need for explanation of the swollen brains. Maybe politics made us what we are (ouch!) 🙂
Indeed, this is not the first time I’ve heard exactly that conclusion. You put it above in much more tasteful terms than “politics”.
Ouch, indeed. Even, YUCK!
MF let me leave perhaps a final thought: Believe me, I understand the appeal of the performative contradiction approach. I agree that it’s an incredibly strong argument to deploy against someone who denies the action axiom. Yes, someone who tries to disprove the action axiom is acting, and in that sense is engaged in contradiction just as surely as Barack Obama saying “no US president is capable of speech.
However, if I show you some subtle contradiction within arithmetic, that doesn’t violate the action axiom. Depending on the contradiction, it might not even violate anybody’s intuitive notions of common sense, at least not directly, in the sense of “A and not-A can’t be simultaneously true.
I guess I wasn’t as clear as I thought I was.
My argument is not that mathematical axioms, that are freely floating and abstracted away from action, will never be exposed as having a contradiction. I think there are contradictions in “higher math” all over the place. It’s why mathematicians now conceive of various independent “logical systems”, such as higher order logic, which mathematicians know aren’t well behaved, meaning they are prone to contradictions.
My actual argument is that I can know that IF one does expose a contradiction in mathematics, it won’t be even possible to do that UNLESS the person who claims the contradiction exists is presupposing that mathematics is logical, and that the error is due to the choice of axioms which I will argue is only possible if the axioms remain abstracted away from action. Since (useful, valid, etc) mathematics is grounded in action, then such mathematics will never be refuted.
Are mistakes made? Certainly. But the presence of mistakes does not mean that nothing can be known with apodictic certainty.
Since it seems you understand a lot about higher mathematics, MF, let me ask: Are you familiar with the Axiom of Choice, and the freaky stuff that it implies? That’s the kind of stuff I mean. You can pick some seemingly innocuous axioms in a formal, deductive system, and generate crazy implications that you didn’t expect.
It’s great that you bring that up, because it is a perfect example of what I was saying above regarding the craziness of “infinity.”
For example, if we suppose the collection of all non-empty subsets of the real line, it is not clear how to find a choice function. Why? Because right away we’re in the craziness of infinity in the true sense of the word. Because we can’t act in choosing among infinite alternatives as means (this gets us into the “deliberation” aspect of action), so too can we not act in choose a suitable choice function for an infinite set of subsets containing infinite elements.
Contrast that with counting based arithmetic, such as the infinite set of subsets that contain, say, the natural numbers, and we can just stipulate a choice function as “the smallest integer in each set.” This is actually a finite based mathematics, even though there are an infinite number of subsets. We are just conceiving a choice for one set and repeating the same choice in a counting fashion, 1, 2, 3, etc, but then we just say “and so on to infinity”, which is really the number 4 when based on action. We can’t act according to a world of infinity. We can’t act by deliberating among infinite means. I think that is why the axiom of choice can get all screwy. Once you abstract away from action, once you introduce the concept of infinity (which is really in the same conceptual sphere as God), then our ability to act breaks down, and hence our ability to do math breaks down.
In practise, that’s why we see iterative methods like Newton’s method usually being taught to 3 or 4 iterations, then he says “this is a good enough approximation.” Then that’s it for the professor’s actions. If we use a computer, then it’s still a finite set of iterations.
So just because our attempt to formalize even simple arithmetic would seem straightforward, it’s conceivable that the axioms actually allow for a contradiction. If this were to happen, it wouldn’t render action itself nonsensical.
I will say it again, if you are going to consider yourself or someone else as being in a position of knowing a “contradiction” exists in a set of axioms, then the person is, consciously or not, grounding mathematics on self-reflective action. The whole meaning of contradiction, the very act of claiming contradiction, means that whatever set of axioms are being considered, the person claiming the contradiction is grounding those axioms back to action. There is just no other explanation. The meaning of contradiction itself carries with it the implication that the mathematics considered are not correct. But not correct based on what? It can’t be the mathematics alone, because mathematics cannot prove or disprove itself (enter Godel). It has to be the case that the person claiming the contradiction is, knowingly or not, telling the world that mathematics is a praxeological, deductive field of inquiry. Making contradictions and identifying contradictions are only possible with an actor present.
And finally: I don’t see what reason the atheist has, for thinking such a world is impossible. Yes, math “seems impregnable” to your mind right now, just like it really seems like there is water in the desert to your eyes when you see an oasis.
Oh come on, that’s not a good analogy. If you’re going to claim, even in a thought experiment, that an oasis is an illusion, then right away you are presupposing knowledge of that which illusions are distinguished from, namely, real water sources. That you know it is possible to “see” something that really isn’t there, logically implies that you know what it means to see something that really is there.
Back the mathematics then, I don’t think you truly get the implications of you saying “it is possible for mathematics to contain a contradiction.” By telling me that you think it’s possible, you are telling me that you think it’s possible for humans to identify something that really is the case, and is not just an illusion.
Let me put it this way: if you’re going to use the “optical illusion” route to cast doubt on my certainty reagrding praxeologically grounded mathematics, then how can you even expect me to know that your exposition of such a contradiction really is a contradiction, and not just another illusion in a world of illusions? Do you expect me to take your exposition in showing such an alleged contradiction to be real? To be true? To be what really is the case? If so, if you expect me to take you seriously and to take your arguments as “true” and “the real deal”, then why aren’t you taking me seriously when I say that praxeologically grounded mathematics is “true” and is “the real deal”? Why bring up the illusion analogy when considering my apodictic certainty, but not your own when it comes to how you clearly expect me to take what you say? You expect me to take you seriously, and yet you are telling me that I can’t take me seriously.
You’re coming VERY close to being a philosophical con artist, who shouts at others that THEY cannot know truth, that they should question their own conception of truth, but they are supposed to take your words as truth. You cast doubt on human epistemology, and yet you put yourself into a position of being the standard, even though you are human as well.
If you question my certainty as being a possible illusion, then how can you expect me to take your certainty of there being a contradiction as not itself an illusion of your doing, or even mine for that matter? If you tell me I’m seeing illusions, then clearly you can’t expect me to see through the illusions at the reality and correctness of what you’re saying.
Why would you think evolution gave you a brain that could discern objective truths about reality, as opposed to approximations that are good enough to outwit a lion?
Don’t look now Murphy, but you just self-obliterated your entire religious worldview. Do I have your permission in using this gem whenever you make any claims to knowing the nature and intentions of God? Or are you going to claim that only you are entitled to claiming objective truth because you have faith in God, and that magic turns your fleshy brain into a truth telling machine that we must all bow under in awe and wonder?
To answer your question: Evolution has such overwhelming evidence the world over, and when I mean overwhelming evidence, I mean there has been exactly ZERO instances of evolutionary theory being falsified, anywhere in the world, for any period on historical record, ever, and not only that, but evolution has also been confirmed experimentally, like we can actually evolve animals ourselves by selectively breeding desired traits (just witness the fact that humans have domesticated wolves over a period of many thousands of years and thus generated an entire diverse population of dog breeds, and in Russia, they domesticated and selectively breeded foxes over a period of 80 years, and the results were that originally hostile, long tailed, long snouted foxes turned into peaceful, short snouted, short tailed, floppy eared, dog like animals).
Archeologists have confirmed via carbon dating the evolution of human ancestors, and have shown that modern humans evolved something like 250,000 years ago from a common ancestor we share along with chimps and gorillas.
How can evolution result in a physical body being conscious of itself, of having the ability to know objective knowledge? I have absolutely no idea, but I know I can do it. I don’t have to know where a computer was manufactured or how it was manufactured, to know it’s a computer.
That I don’t know doesn’t mean there exists God.
I can imagine that IF physical matter is going to become conscious of itself, and have the ability to reason and learn objective reality, then the first entity to do that through natural selection will be alone. As other like animals begin to be born, and they too develop the skill, I would imagine that this race of animals would almost certainly become the dominant animal on the planet they arise.
If I ask you where objective knowledge comes from, and you say God, then I can ask how you can have objective knowledge regarding mathematics merely by virtue of believing in God. You have no idea why there is a God, when you try to explain what you don’t know by saying “God did it”, you’re really just saying “I don’t know” in another way.
No matter what theory you have regarding creation, your acting physical body isn’t going to transcend its own nature. You’ll still be subject to the same Earthly laws as everyone else. It doesn’t take an atheist to know this.
Let me turn this around: Suppose that humans discover a more intelligent alien race that looks nothing like us. Would that falsify the doctrine that God created man in his own image? Suppose they wiped out the entire human race. Would that falsify the bible because it wasn’t Jesus that brought revelation, but an alien race? Or would you say that by “Jesus”, the writers of the bible really meant the leader of the alien race?
Just consider the entire universe, and the possible alien races. There are probably tens of billions of intelligent races out there. Oh what crazy megalomaniacal mentality is required to think of oneself as made in the universal God’s image, and not only that, but to claim to know it, and not only that, but to be only a portion of the human race that actually believes it!
You may think atheists are being arrogant and self-assured and all the rest, but I can tell you that we atheists are the ones who are truly in awe at how arrogant theists truly are, how utterly hypocritical it is for you guys to even hint at labelling non-theists as arrogant and self-assured, and then to add insult to injury, to belittle them for daring to claim to knowing objective truth out your front door, after which you go out through your back door and preach how you are able to know objective truth while you religionify.
For a theist to use the tactic of encouraging skepticism in his opponents, is truly the most jaw dropping hypocrisy ever. Just look at what you guys are actually claiming to know. You are claiming to know something responsible for the entire universe, you are claiming to know the intentions of that universal creative force, you are claiming to know the origins of everything, you’re claiming to know of the existence of magical places called heaven and hell, you’re claiming to know that the secret of human happiness and human prosperity is to believe in an invisible man in the sky, and you’re telling atheists to be more humble? I don’t know what else to say other than wow.
I subscribe to that, in full. It does not mean, though, that I don’t love my fellow human beings, even theists!
😉
8) 8)
Mathematicians DO find provable contradictions in mathematics, but when they do, they change their assumptions to get rid of them. Read the history, e.g., of the development of axiomatic set theories. It is an emergent order with a strict prohibition against provable contradictions. Separate mathematical systems can be set up that contradict each other, but by themselves lack any proven contradictions.
And contradictions are prohibited, because a contradiction IS a falsehood–a failure of understanding, where understanding is the very goal and purpose of human thought.
Mathematics, you would expect, always has an unprovable aspect, since ultimately it is a tool built to understand the merely existing observable world. That is, you observe adding a sheep to your flock, or a mile to your road, or a cow to your heard. That isn’t a statement, but an observation. It just is. Your mind then separates out the different aspects of the observations and identifies the commonality of x+1 (or in set theory notation, X U {x}). You discover that what holds true of this abstraction holds true of any other observation so abstractable, such as adding a child to your family. If you ever find out it doesn’t work (contradiction), then you make the necessary changes until it does, because what you want is something that works.
Perhaps someone could see something mystical in the fact that anything at all can be understood, or be constant (although if that were not the case, then nobody could ever see anything), but mathematics is truly mundane. Beautiful, interesting, complex? Yes. But inspiring mystical assertions? I don’t see it. There just isn’t so much profound mystery in it as so many people throughout the ages seem to think. For any beautiful mathematical presentation of shear genius, there is a definite unbroken identifiable chain of uninspiring common logic ending in the most basic of abstracted perceptions. 100% of its parts and their connections are without mystery.
“Mathematicians DO find provable contradictions in mathematics, but when they do, they change their assumptions to get rid of them. Read the history, e.g., of the development of axiomatic set theories. It is an emergent order with a strict prohibition against provable contradictions. Separate mathematical systems can be set up that contradict each other, but by themselves lack any proven contradictions.”
“And contradictions are prohibited, because a contradiction IS a falsehood–a failure of understanding, where understanding is the very goal and purpose of human thought.”
vikingvista, you say that a contradiction is prohibited because a contradiction is a falsehood, and all falsehoods should be rejected.
Question to you: What would you say are the grounds for us knowing that a falsehood or truthhood is even present, and what is the ethical foundation for saying that falsehoods should be rejected as opposed to merely tolerated, or even accepted?
“What would you say are the grounds for us knowing that a falsehood or truthhood is even present,”
The grounds are the purpose of the person engaging in these abstractions. Imagine you are working out a business plan for acquiring Bill’s sheep. If you are of the opinion after your time experiencing and thinking about the world that Bill’s flock can only be either empty or not empty, and never both, then a result of your reasoning that concludes that Bill’s flock will be simultaneously both empty and not empty simply doesn’t work for you. You must reconsider your reasoning.
And if you are like most people since Aristotle, the idea that anything both is and is not, is incomprehensible, let alone observable, and therefore forms no part of anything meaningful to you. The law of noncontradiction is therefore not only a useful assumption, but a necessary one, for a practical life–nothing you can ever observe or conceive will be a true contradiction (simultaneity of being and not being), so for your reasoning to match your knowable reality, with knowable reality being your concern and purpose for thought, contradictions are rejected.
(Once a formalism is symbolized, one can of course play with this formalism in ways that formally accept this inconceivable state of affairs, as with a so-called paraconsistent logic, but then the purpose of the thinker is different, and my grounds-are-the-purpose answer still holds).
“and what is the ethical foundation for saying that falsehoods should be rejected as opposed to merely tolerated, or even accepted?”
Ethics has nothing to do with it. Accept contradictions if you want. I can tell you that your calculations will not be meaningful in the end, but that doesn’t mean you are being unethical.
But for someone who wants their conclusions to land firmly in the realm of the conceivable, contradictions and all other inconceivables are rejected. The realm of the conceivable is a very practical place to be, which is why I say mathematics is mundane. There is nothing mysterious or mystical at all about it. The assumptions and steps are all there, widely grasped, and chosen for a purpose. If there is a god in mathematics, then he must be in the assumptions, the very process and structure of thought, or in the observations from which the quantitative abstractions are formed. But all of those things are quite different than the inspiring beauty of mathematics being discussed here.