Monday, February 20, 2006

Stupid Spin and False Advertising

I normally enjoy posts in the blog "On the Fence": but I don't enjoy posts that just get it wrong.

And J. Kelly Nestruck just gets it flatly wrong in the post "Math joins the postmodern world".

Let us start with the title, which suggests a level of relativism few mathematicians would buy into (I speak as a reformed one, now working on other things, but always caring about mathematics). The sole evidence for the post is a New Scientist article titled, "Mathematical proofs getting harder to verify". The substance of the article does not go beyond asserting that exhaustive searches in some proofs are being handed off to computers - and the proofs include the algorithms used. This is, to start with, old news, and actually a pretty incremental deivation from mathematical history. Proofs always need verification, and it can fail for all sorts of reasons; there is no major change in principle in allowing the refutation of a computer search to refute a purported proof.

Postmodern? How? This just seems silly and glib. Does Nestruck have the vaguest idea what this all really means?


At 3:25 PM, Blogger BensonBear said...

What do you think of Hersch's "What is mathematics, really?" or Lackoff's "Where mathematics comes from"?

At 5:47 PM, Blogger Alan Adamson said...

No idea. Have not read that sort of stuff in 20 years. Might start doing it again when I retire. I did just do a scan of the reviews and discussions of these books and saw nothing new compared to 20 years ago when I cared a lot about this. None of it seems to bear in any significant way on this silly idea of postmodernism as somehow impinging on current understandings of mathematics. Mathematicians almost always experience their work as Platonists, which is in no way to suggest they need such a clear-cut philosophy in the end. But the use of search eliminations by computer in proofs is no major step. It is harder to check, but no principles change. Exactly as the original article said, which in no way invited broader inferences.

At 11:54 PM, Blogger BensonBear said...

Oh, I think those books do bear on "postmodernism", but if you don't currently care about those issues, not much point elaborating.

At 9:24 AM, Blogger Alan Adamson said...

Well if postmodernism remains a fashion when I reach retirement, perhaps I may choose to read them. But the original article that started this chain had nothing to do with postmodernism, simply with new tools being used in mathematical proofs. And I've never found much content in what I've consumed so far as 'postmodernism'.

At 2:44 PM, Blogger Martin said...

I like Bensonbear. But I am struck by your resolution not to consult his suggested readings (that said I know that few read as widely and as much as you do so you probvably dont need reading lists).

But at the risk of coring the context of postmodernism here the idea encapsulated in the question: is mathematics discovered or constructed? Because if that is tru then of course little will have changed in 20 years. Little has changed since Plato in that regard although one hopes we can write more about it all.

I'll risk suggesting a book to you to: Hilary Lawson wrote a book called "Closure: a theory of everything" which I found interesting and perhaps the first arealist description of science.

At 4:35 PM, Blogger Alan Adamson said...

I'd say constructed in a very disciplined way. But I don't think that means the outcomes are arbitrary. Lawson's book sounds interesting - I shall look out for it. I have nothing against any of the books suggested - I'm just not terribly interested in the philosophy of mathematics at the moment.

At 3:26 AM, Blogger BensonBear said...

Despite the lack of interest exhibited by our host, because of his technical expertise in these matters I can't resist commenting a little on the question that Martin asks.

Even if mathematicas is "constructed", we agree that it is not constructed "arbitarily" but it a "disciplined" manner. Indeed.

Goedel in his 1951 Gibbs lecture observes that "the activity of the mathematician shows very little of the freedom a creator should enjoy. Even if, for example, the axioms about integers were a free invention, still it must be admitted that the mathematician, after he has imagined the first few properties of his objects, is at an end with his creative ability, and he is not in a position also to create the validity of the theorems at his will... That which restricts [the freedom of creation] must evidently exist independently of the creation".

If this is right, and I believe it is, I further believe that that which exists independently of the mathematical creation (such as it is) can in effect serve as the ultimate mathematical reality, at least some part of it, a part that is utterly independent of both human biology and social forces.

How this relates to postmodernism and the books I mentioned? Well, one very strong feature of the postmodern is to deny the universal and also to deny reason as something that extends beyond biology and social forces. (Ironically, in this sense a very strong naturalistic scientism is actually very useful to the postmodernist, while at the same time of course he wants to deconstruct it).

The link with the books is that, as I understand it anyway, Lakoff and Nunez take the first tack -- mathematics is grounded in human biology, and Hersch takes the second tack -- mathematics is grounded in society (and he even distinguishes between left and right wing mathematics, opting of course for "humanistic" left wing mathematics).

Incidentally I looked at Lawson's book in the past and I am afraid I concur mostly with the comments at Amazon (including the counter-recommendation of Thomas Nagel's "The View from Nowhere")

At 7:45 AM, Blogger Alan Adamson said...

Bensonbear, you are waking dormant demons. :-) And certainly raising the tone of discussion on my blog! Thanks!

Goedel was a strong Platonist, who believed the subject matter of mathematics was 'out there', which is ironical, given the ways his incompeteness theorem is cited often to support relativistic views. I have always considered it a bulwark in the Platonist defence. When I actually 'did' mathematics, the experience was profoundly Platonist - one was trying to trick the world into showing some new side. I certainly always understood that the body of known mathematics was constructed by the vast community of us over time who had worked on it. This is not to say it was simply invented in some arbitrary manner. And I find it hard to believe, knowing the rigorous discipline on what gets counted as interesting and valuable content, that this body of 'literature' (I am being a bit provocative) that is mathematics, is very dependent on our biology or our 'social context' or 'politics'.
I think what troubles me about much of what I read from 'postmodernists' is that there is a poor appreciation of how hard it is to 'know' something, and how valuable. You point out the fundamental contradiction in the positions they espouse, and this is one reason I expect the whole movement will just go away in due course.
All I see in it is a skepticism less sophisticated than Hume's, and then claims of authority based, given their skepticism, on nothing of any importance.

At 4:43 PM, Blogger BensonBear said...

I'm leaving aside the issue of postmodernism, but I was hoping that you would reply to my (perhaps too implicit) criticism that the idea that mathematics is "constructed" is self-defeating, since its "construction". in many cases at least, amounts to no more that specifying an "in" to previously existing structures, those that serve to constrain subsequent construction.

To be concrete, once the axioms of Peano Arithemetic are specified, it is a hard-core fact, totally independent of our whims or biology, that they are either consistent or not (Is it not?) The first problem for a realist, that apparently is not faced by a Lakoff or Hersch (perhaps though because they have their heads in the sand), is the metaphysical issue of what the actual truthmakers for this fact are. (It refers itself to an actual infinity of proofs, saying that none of them is a proof of a contradiction from the axioms). I think a realist here has to be a pretty strong realist: Aristotelean realism will not do.
That seems pretty unsavory for the modern "scientifically" minded person.

The second problem is the epistemological one. You seem to like evolution. But it seems hard to explain how evolved organisms that fit into a finite world can manage to even refer to such infinite structures, let alone think true thoughts about them.

On what grounds do we think PA is consistent? Surely it is not empirical grounds? So it is a clear and distinct idea? How is that reconciled with evolutionary naturalism?

At 4:54 PM, Blogger Alan Adamson said...

I'll follow you a bit of the way but not very far - as I said, I don't find this really an attractive activity.
I think there is enormous circumstantial evidence (the best our limited systems will ever get) that PA is consistent. We have not found a problem with it. Moreover, we have managed to achieve a great deal on its basis (yes much engineering depends on this stuff, and that engineering works robustly).
You will have to explain Aristotelian realism vs. strong realism to me - I have no idea what the distinction is.
And yes I agree hte reason Lakoff and Hersh don't have this problem is that their whole universe has vansihed under them and there is no longer any basis for discussion.
I am afraid I do not see what any of this has to do with theories of evolution.

At 5:04 PM, Blogger Alan Adamson said...

What's the problem with finite creature conceiving and understanding infinite notions? Is that not part of what things like the Peano axioms are for?
Why should we not accept that our knowledge is always partial?
What is the technical problem you see?

At 3:54 AM, Blogger BensonBear said...

I don't see a technical problem, in the sense I originally referred to your expertise as being technical (my belief was and is that technical knowledge could be helpful in examining these issues). But I guess I do have a problem seeing how our knowledge about mathematics can be partial, putting aside the trivial facts that we don't have enough time to figure it all out, and that we often make errors in reasoning.

In the case of things in the natural empirical world, we can see why at a given time our knowledge of fundamental law might be partial. We simply have not yet acquired enough data about what is out there. But in the case of mathematics (the sort I have referred to so far, about nothing but first order logic and the natural numbers), it seems that, in a sense, all the data is completely in. Numbers are not about our bodies, our society, or the natural world. If we don't know everything about them now (again, modulo time and error), then what else are we going to be able to do that will tell us something we do not already know? We can reason (and run programs to help us), but what other source of data could there be? Surely to say something like "mathematical intuition" is not much of a solution, and further, it is pretty hard to identify where in a material, evolved mechanism that intuition can be located.

A la Dennett, it may be quite possible to tell a story about how a mechanism could evolve such that it contains in it various syntactic objects which operate in a way quite similar to our own syntactic productions about numbers and logical systems and the like. But nothing about this naturalistic understanding would seem to justify or even required realist interpretation of these sentences, an interpretation which I am very strongly disposed to accept. Perhaps Dennett could also explain this strong disposition as an illusion, in a similar manner to the way he "explains" consciousness. In fact, I think the two (hetero?)-phenomena are quite similar and perhaps even actually related: I assert that I have a clear and distinct idea that, for example, every well formed statement about the integers is either true or false because I can see this clearly in my mind's eye. The phenomenological aspect of this seems extremely important (interesting that Goedel got very interested in Husserl in later life), so that a mechanism lacking in consciousness, although banging the table about this as loudly as I might, would nevertheless seem to be lacking, in more ways than one.

"Seems" would be the operative word for Dennett. But I don't buy his account of consciousness (it only seems like we are conscious!?) and I don't think I could buy this sort of account about mathematics either. Maybe this will be his next book. After dismissing consciousness, teleology, libertarianism, and now religion, what else could be left?

At 11:50 PM, Blogger J. Kelly said...

I was, yes, mostly being glib.

At 6:13 PM, Blogger Martin said...

It may be too late for me to usefully add this comment (I love the punctuation provided by the last comment) but I repeat my recommendation of Lawson to you. It is probably my deep ignorance of the "real literature" in this area that allows me to see this book as somehow clarifying but I nevertheless did.



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