The Empirical Justification of Mathematics

Richard Brown 22 Comments

Things have been quite around here lately, mostly because I have been lost in GTA IV and finishing my dissertation. Well I am making some progress (on both 🙂 ) and so will try to get to some of the comments around here.

In answering the comments on the Refutatioin of Rationalism I started thinking about Quine’s indispensibility argument for the empirical justification of mathematics. The argument starts from Quine’s claim that we are ontologically committed to the things which we quantify over in our best theories. The indispensibility of mathematics to physics means that we are committed to the existence of numbers (but not, obviously, to their non-physical existence). So Quine went on to argue that, since our theories all get confirmed or disconfirmed together as a group, the empirical confirmation of physics is empirical confirmation for mathematics. In this way mathematics is empirically justified.

One problem with this argument is that it depends on confirmation holism. That is, it depends on the claim that all of our theories are confirmed or discomfirmed together. None ‘face the tribunal of experience alone’. I then started thinking about how Rosenthal’s version of this argument avoids this commitment and so is a better argument. Sadly Rosenthal has never published this argument (I heard it in a seminar on Quiene and Sellers he gave) so I will try to recreate it as best as I remember.

The basic idea is: if we ever had empirical evidence that some truth of arithmatic was false we would have to admit that it was false. But if so then mathematics is empirically justified. To make the case he asks us to entertain the following scenerio. Suppose that you had two pens of sheep; one with 6 and one with 7 sheep. Now suppose that you counted the sheep individually in each pen (and got 6 and 7) and then you counted all of the sheep and got 14. Suppose you did it again. 1. 2. 3. 4. 5. 6. Yep six sheep in that pen. 1. 2. 3. 4. 5. 6. 7. Yep seven sheep in that pen. Then all the sheep. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. Suppose that this was repeated by all of your friends with the same results. Suppose that it was on the news and tested scientifically and confirmed. Suppose that this phenomenon was wide spread, observable, and repeatable.

If this were the case we would be forced to admit that 7+6=14 is true therefore mathematics is empirically justified.  

The Refutation of Rationalism

Richard Brown 19 Comments

I am back from my vacation in Liberty City 🙂

Seriously though, gta 4 is a lot of fun and very addictive!

I have recently been very interested in showing that rationalism is hopeless. To be open, I confess that I used to be a rationalist when I was younger. When I first started studying philosophy I was very influenced by Descartes and found his talk of clear and distinct ideas and ‘the light of nature’ very compelling. There is no suprise here, as rationalism has been the dominant view in the Western tradition since its inception. But, as rationalists like Bonjour admit, rationalism has suffered several notorious and embarrassing setbacks. Perhaps the first of which goes back to Gallileo showing empirically that Aristotle’s physics was wrong in assuming that heaverier bodies fall faster than lighter bodies. More dramtically, perhaps, is the discovery of non-Euclidean geometry which showed that various of the fundamental postulates of geometry were not necessary (like the parralel postulate). We should perhaps add to this the discovery of Russell’s paradox and the various self-referential semantic/syntactic problems (i.e. Godel, the liar, etc) which have led to the development of alternative logics.

What this shows is that intellectual seemings are fallible. It cannot therefore be argued that something seeming to need no empirical support and seeming to be justified purely by reason is enough to establish that the fact in question is really justified independently of experience or not. So, take 1+1=2. It certainly seems that this is true, in fact it is hard for me to imagine how it could be otherwise. There is a strong subjective sense of certainty that I experience when I think about it. All of this is no doubt true. But we have as of yet no reason to think that it is REALLY necessary, or that its justification is independant of experience. This is because we can not tell a priori whether the intelectual seeming is indeed correct. This shows that rationalism is in serious trouble. There is no other reason to take rationalism seriously other than the strong pull these rational insights have on us.

One might want to reply by saying that is overly skeptical. We shouldn’t abandon a priori knowledge just because we have mistakenly identified somethings as necessary which weren’t. So too, the objector goes on, just because we hallucinate doesn’t mean that we don’t normally see objects. Fair enough. But then what we need is an actual account to back this up. What is the difference between the cases? We can give that in the empirical case. We can describe ways in which we could find out whether the person was hallucinating or not based on our ability to monitor the brain of the animal in question and our visual impressions of the experimental set up. We can give a sketch, if not every detail, of a story which desribes how the brain interacts with the enviornment it finds itself in and generates representations of that environment. But can you do the same for rationalism? To date no one has. What is an eternal, necessary, non-physical/non-natural object like a number or modus ponens really like? How do we interact with it? No one knows. How could they?

Now this would be a pressing concern if it were impossible for us to fully understand the world we live in except for the truth of rationalism. But this certainly isn’t the case. We have good candidates for materialistic accounts of every disputed area. For instance, in the area I know most about, we have the mind-brain identity theory and the higher-order theory of consciousness. I do not mean to say that we know that they are true, but only that they are viable candidates. For all we know right know they could be true. They have not been absolutely refuted by any a priori arguments, nor have they shown themselves to be inconsistent with the findings of science, quite the converse actually. As for math and logic we have either constructivism or Mill’s view that they are empirical generalizations (I interpret this to mean that they are an attempt to model the way that the physical world works and to grow into Quine’s indispensibility argument that the justification for mathematic is empirical), or a more modern version of deflationism or fictionalism about this stuff. The same is true for ethics.

Again, none of these has been demonstrated to be correct. The point, rather, is that we should prefer natuaristic/materialistic accounts ove their non-natural/non-physical competitors. They automatically become more plausible because of their reliance on the more plausible empiricist/scientific account of knowing.

Einstein and the a Priori

Richard Brown 5 Comments

Tanasije has recently offered up Einstein as an example of how empirical science is dependent on a priori knowledge. His point seems to be that once Einstein had his two fundamental principles (i.e. the principle of relativity and the principle of the constancy of the speed of light) he was able to use ‘pure reason’ and deduce a priori the rest of his physics. But the question of a priori knowledge, I think, is the question of the status of the first principles and the status of the rules that we use to deduce the rest of the physics. It is only if you think that the two principles + rules for deduction are known to be necessary facts about the nature of reality independently of experience and solely by reason.

The empiricist can account for what Einstein did; the two principles followed from empirically validated theories of the time (the principle of relativity is stated by Galileo and is a part of Newtonian physics as Einstein knew it and the constancy of light is predicted by Maxwell’s equations). The rules that are used to deduce (i.e. classical logic) are highly useful empirical generalizations. So Einstein does indeed start from first principles and deduce physics (if that is indeed what happened), but nothing in the story told by the empiricist is really independent of experience.

In order to have some serious rationalism going on, one would have to add to Tanasije’s accoount the claim that Einstein’s principles and the rules of classical logic are known by reason to be necessary facts about reality. Einstein does seem to cite a thought experiment as evidence for the constancy of the speed of light. He says he imagines chasing after a beam of light. What would it look like? If Einstein could catch up with the beam and look at it how would see the light standing still, which is unintuitively odd. So, he must not actually be able to catch up to the beam. But the only way that that is possible is if its speed were constant relative to Einstein no matter how fast he went. QED.

But is this really evidence for rationalism? Not quite. Some scientists (Paul Davies, for instance) think that that the speed of light may have been slowing down since the Big Bang. What is going on here seems to be this. We have a theory wich is the one that best unites disperate phenomena and is empirically adequate. We usually have outlying data and often scientists take creative leaps to integrate these outlying data points and thereby unite more disperate phenomena and provide greater empirical adequacy. It is plausible to think that Einstein himself was motivated by a conflict between theories he found intuitively compelling for the reasons cited above.

The moral of the story? Intuitions are theory driven and not ‘tother way ’round!

Not as a Means Only

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In an earlier post I argued that the categorical imperative entails that we are not allowed to use sentient beings as a means only. The semantic Terrorist challenged me on this (and other things) and argued that I couldn’t really mean it. But I did, and I still do. But what does it mean to not treat something as a means only?

Let’s start with the human case. Does the categorical imperative entail that I cannot ride the bus? Afterall, don’t I use the bus driver as a means when I ride it to work or the club or whatever? Not necessarily. I certainly treat him as a means, but I needn’t treat him as a means only. So, what the categorical imperative entails is that I recognize that he is an individual. This requires no more than a friendly smile, or greeting, and perhaps a ‘thank you’ upon exiting. What this will actually mean in detail will depend on the actual relationships that one has to these individuals. So, one has a duty to treat ones wife as an end in itself, and so too with the bank teller. But what it will actually end up meaning is quite different because of the relationship that one has. So there will be ways that you could treat the bank teller that might count as respecting his being an end in himself, but those same actions would not count as treating your wife as an end in herself, and vice versa.

It seems to me that this general idea can be extended to animals. We have a duty not to treat them as means only, which clearly rules out the way that we treat them now.  But we can treats them as means, but we are obligated to do so in a way that respects their sentient capacities. So, we are not morally required not to use the donkey or horse as a pack animal but we are obligated to recognize that the animal deserves to be treated with respect and in a way that does not cause it unecessary harm or suffering. It also seems that what ones duties are depends on the relation one has to the animal. So, one may be obligated to treat ones pet in ways that do not come up with respect to wild animals (that is, beyond the basics due to all sentient beings).

The A Priori Argument against Rationalism

Richard Brown 4 Comments

In a series of earlier posts I have been giving an evolutionary argument against rationalism. In the course of doing so I have appealed to Devitt’s abduction argument. But I also think we can give an a priori argument against rationalism. That is, an argument to the effect that the empirical method is the only way to know anything about reality.

Is it conceivable that there are absolutely no necessary truths? It would seem not. For that would mean that it was true in every possible world that there were no necessary truths but if that were the case then it would be a necessary truth that there are no necessary truths (since it is true in all possible worlds) and that would make it false that in every possible world that there are no necessary truths. So we derrive a contradiction from the assumption that every truth is contingent and so must conclude that there is at least one necessary truth.

But is it contradictory to think that there were only one necessary truth? Could it be necessary that everything (other than this statement) was contingent? My intuition tells me that it is a neccesaary fact about reality that everything (besides this fact) is contingent. If you take the time to clearly and distinctly think about matters you will come to see, by the light of reason, that everything is necessarily such that it is contingent. This is because each thing which is claimed to exist of necessity can be conceived not to be necessary. And the claim that it is either necessary that p or necessary that not p can be conceived to be false as well, as when I conceive that numbers exist as non-physical objects in some possible worlds and do not exist at all in others.

According to the rationalist it is conceivable, so possible, that there be just this one necessary truth. Since it is possible that it is necessary, it is necessary.   So we can have a priori knowledge about the world, but it consists in this only: we must know the world via the empirical method.

Valid but not in Virtue of Form?

Richard Brown 1 Comment

Some remarks of the Semantic Terrorist in the post on moral truthmakers got me to thinking. Here is what he said.

consider the following argument which is easily proven to be invalid despite the fact that many analytic philosophers would mistakenly classify it as valid:

1) George Bush is a bachelor.
? George Bush is unmarried.

This argument is in the same logical form as the following:

1) George Bush is a Texan.
? George Bush is unmarried.

As ST points out many analytic philosophers do take that argument to valid. It is standard in logic textbooks to point out that these arguments meet the definition of validity; it is indeed impossible for the premise to be true and the conclusion to be false, but as the point continues, this isn’t because of teh form of the argument. The form, as ST demonstrates, allows for counter-examples, and so the validity must be due to something besides the form of the argument. The reason for the impossibility of the truth of the poremises and the conjunction of the denial of the concusion is said to be due to the definition of ‘bachelor’.

All of this is standard, but why isn’t the argument above seen as having a supressed premise of the form ‘all bachelors are unmarried makes’? Then the argument is formally valid; it is just an instance of a very common categorical syllogism. The same is true of the texan argument, it just happens to have a false suppressed premise ‘all Texans are unmarried males’. I don’t see what the argument against positing the suppressed premise is supposed to be. It is clearly the only way to make the inference legitimate.

Empiricism as the Default Position

Richard Brown 21 Comments

In the comments on the previous post Brandon says

You seem to be assuming that we all start from the empiricist side, and then the question is just: why should we go farther, into rationalist territory? But, of course, most rationalists won’t concede that empiricism is, or even can be, the default position

I was thinking about my response and then I happend to read this review of a new book on Quine. The review portrays the book as an attempt to show that Quine was a systematic philosopher who was trying to give a completely naturalistic account of the world (what the author call naturalism, I would call empiricism but that is probably just terminological).

According to Quine, the fact that the naturalistic outlook of science is the best option at our disposal is itself discovered empirically by means of scientific criteria of evaluation: in fact science has given us the most comprehensive, systematic, successful and intelligible account of the world.

I absolutely agree with this interpretation of Quine and also with the claim that the questionable parts of Quine’s philosophy (like the behavorism) are Quine demonstrating that we could in principle give naturalistic accounts of phenomena that the rationalist claimed only they could do. We have better accounts now, but the Devitt’s adbduction is very much in the spirit of this interpretation of Quine.

Empiricism works, in principle, just as well as rationalism. Anything that can be accounted for by rationalism can be accounted for by the empiricist. The empiricist appeals to entities and processes that are (fairly) well-understood. The rationalist appeals to entities and processes that are completely mysterious. No one has ever even attempted to seriously give an account of what we know and how on a rationalist conception. On top of this empiricism has itself been tested by its own standard and passed. If there were some compelling reason to think that only rationalism could account for some phenomena then I agree that would be good reason to opt for it but when there is an empiricist alternative it should be preferred.

 

The Evolutionary Argument against Rationalism

Richard Brown 13 Comments

In the comments on my post on Armstrong on Naturalism and Empiricism Brandon and JS raised a concern about the evolutionary argument I sketched against rationalism. I have hinted at this kind of argument before but I guess I have never spelled it out. I have sort of thought it was just too obviously an extension of Hume’s kind of argument…but maybe it is worth spelling out.

Suppose that you have a stable environment E. In E there are events (e1, e2,…en) and it so happens that there is a regularity in E such that e2 regularly and reliably follows e1. Now suppose that there are creatures, C,  that live in E. Then evolutionary theory says (roughly)  that C will adapt to E via natural selection (I do not mean to be saying that C will become optimally adapted, that is a hotly contested claim). One obvious source of reproductive advantage would be being able to track the regularities in E. If e2 is such that it is (harmful or) beneficial for C then it would, through the course of natural selection, come to be the case that C  tracked the regularity, R, expressed by ‘e1 then e2’; that is to say C would come to associate e2 with the occurrence of e1. Now assume that R is some basic and (hitherto) extremely reliable regularity (like that when you have one object and you place another object next to it you then have two objects).

Now suppose that, for whatever reason, R continues to be regular and tokens of e1 are immediately followed by tokens of e2. Decedents of C will also track R, but will also have to track other regularities that they discover for themselves. So, decedents of C that have R ‘hardwired’ in will free up cognitive resources. So now suppose that C is a species from which a species evolved from which a species evolved…from which we evolved and that R has continued to be regular and reliable up till now. At this point R would presumably be something for which we are equipped to find intuitively obvious. This is exactly what developmental psychology (like work from Spelke and many others) has shown.

So, when the rationalist appeals to their experience of ‘finding something obvious’ as evidence that we know something in a special way that is revealed to be necessary and universal, it could just be the product of that regular and reliable conjunction of events has, around here and so far been that way. But that, of course, does not guarantee that it must be that way. Now how could the rationalist know which of the two theories better accounted for his experience? Our intuitions are shaped by the world that we happen to find ourselves in. And so if evolution is true rationalism is most likely false.

JS asks,

I’m also not sure evolution precludes intuition as evidence for rationalism. I suppose I just don’t think your reasoning pans out in real life. I don’t see how I can’t both agree with this: “we would expect that there would be certain truths which seem to be self-evident but are really just the product of a long process of natural selection in response to a stable environment”, and not believe in rationalism. Can’t I tell an equally nice just-so story like this: I agree with RB’s quote, but at a certain stage of complexity it’s been found that animals begin to intuit other actually self-evident beliefs, which then survive natural selection in virtue of their fitness. The end. I’m sure you disagree, however.

Well it would depend on what you mean when you say ‘animals begin to intuit self-evident beliefs’. If you don’t mean ‘discover solely by the use of reason necessary and universal facts about reality’ then we do not really disagree on anything. But if you do mean that, then the question is how could those kind of things be selected for or have any fitness at all if they did not causally interact with C or E? Sure there are spandrels and such, but in order for a property P to be reproductively beneficial it would have to play some role for C or in E. Notice that the mathematical truths play a hugely beneficial role for us. It is in fact their usefulness and indispensability to science that Quine and others have taken to show that they are empirical. David Rosenthal has given (in class) a similar argument for logic being empirical. He argued that one reason to think that it is is that it explains inferences that humans find valid and to that extent is hostage to the empirial–and so not analytic.

But maybe we did evolve to track necessary and universal facts about reality discovered by the use of reason. The point is that we couldn’t appeal to our experience of certainly as evidence for that view since we can’t rule out that the experience isn’t due to a very long association between certain events (that are not necessary). We would need some other kind of evidence. But what other evidence is there?