On the relationship between formalism and empiricism

[ultrafilter]

I found an interesting essay on the relationship between mathematics and physical science. Specifically, the author is deconstructing the marvelous efficacy of mathematics as a tool for modeling physical reality, and claiming that it's not so miraculous after all.

I think it's a reasonable case. Any thoughts?

[Spiritus Mundi]

I think it's a pretty good summary, but the aphorism at the end is a bit too dismissive for my taste. I think the author makes a serious misstep in the sentence: The weather, or the behavior of any economy larger than village size, for example -- systems so chaotically interdependent that exact prediction is effectively impossible (not just in fact but in principle). To illustrate my objection, ask yourself how the principles which he references were derived.

For that matter, when I was in school probability theory was considered a branch of mathematics, so the observation that many systems can only be accurately modeled through probability hardly seems to negate the question, why are mathematical models so successful at predicting real world results.

Even if one accepts the author's restrictive attention to only strictly predictive mathematical models, I don't see how the observation that such models fail to predict everything answers the question posed.

[erislover]

From the link
For example, the positive integers are a good formalism for counting marbles. We can confidently predict that if we put two marbles in a jar, and then put three marbles in a jar, and then empirically associate the set of two marbles with the mathematical entity 2, and likewise associate the set of three marbles with the mathematical entity 3, and then assume that physical aggregation is modeled by +, then the number of marbles in the jar will correspond to the mathematical entity 5.

They will, or that is what we call five? A question worth answering, especially given his following comment on teaching children to count (not quoted in this post).

Much more complicated because, as we now know, there are mutually exclusive ways to axiomatize mathematics!

And how would we clever, persistent apes choose among such axiomatizations on empirical evidence without begging the question?

I don't think apriorism or transcendentalism in mathematics has quite been vanquished. As a side note/question, doesn't the math behind QM largely consider particles as points?

Spiritus, just to be clear on your point about chaotic systems, you are suggesting that in order to show that we cannot model the behavior we must already have accepted a model for behavior whose inputs are too sensitive, or long gone, to be empirically determined, and so we cannot in truth say that exact prediction is impossible in principle? That is, how do we know this is the model that we can't use, rather than just a model we can't use? Is that the question we need to answer to satisfy your objection?

[Ramanujan]

Originally posted by erislover
I don't think apriorism or transcendentalism in mathematics has quite been vanquished. As a side note/question, doesn't the math behind QM largely consider particles as points?

it does, largely. many people think that is one of the major obstacles to a unified field theory.

one of my problems with arguments of this sort stems from the author relieving himself of credibility through his argument. it's the same as when someone claims that god is beyond logic. if logic is our only way to produce convincing arguments, and it's not good enough, you relinquish your ability to discuss the topic. that isn't quite so apparent here, but logic is a pure mathematical system (if we accept russell's definition of pure mathematics). and without logic, we can't say whether an argument is valid.

it's a rather subtle problem, and i don't know if there is a way around it. but it does seem quite pointless to argue that logic is not transcendent, since it is our only tool for assigning validity.

[Spiritus Mundi]

erl
Ask yourself, what principles might one use to determine that a system was too chaotically interdependent to allow precise prediction? It ain't poetry, though I have often wished that I could substitute an astute alliteration for a proof.

[ultrafilter]

Originally posted by Spiritus Mundi
For that matter, when I was in school probability theory was considered a branch of mathematics, so the observation that many systems can only be accurately modeled through probability hardly seems to negate the question, why are mathematical models so successful at predicting real world results.

Consider the statements X = Y and P(X = Y) = p. While they're mathematically the same (two quantities are equal), the predictive power of the first is greater than that of the second.

[erislover]

Ah, principles which were "found" through the suggested feedback mechanism (formalism + empiricism) from the very systems we are now perplexed by. Then the statement could only be true if we did accept apriorism. Yeah?

[Spiritus Mundi]

You got it, erl. Or, at the least, our confidence in the principles is bounded by our confidence in the formalism:empiricism relationship, which makes the author's use of said principles to dismiss said relationship a bit self-defeating.

ultrafilter
No argument, but probabilities also offer a formal structure which makes predictions and functions in a feedback cycle with empirical observation. The author's reliance upon teh latter to dismiss the significance of the former strikes me as arbitrary.

[erislover]

But see, Spiritus, this is exactly my issue with Popper's view on empirical research through falsification. He attacks apriorism like it is a disease (and hey, who wouldn't? ), but relies on assumptions (which are a priori to his theory, not pure a priori assumptions, but of course he never decides whether they are pure or not, and it must be inferred in the negative with respect to his dislike of apriorism) of all sorts of things that will simply "take care of themselves" before one comes to science, after which time we may safely employ the scientific method, which itself is goverened (as he describes it) by a sort of feedback loop much like the one described here—predictive theories are given no metaphysical weight (a way to look at formalism, to be sure), but their predictions are tested against evidence. Now, Popper wants this to be the logic of scientific discovery, but I see him present nothing as to why it shouldn't be the logic of empirical discovery.

In the book (The Logic of Scientific Discovery) he even mentions (more than once) that he is not, for example, putting forth a theory of meaning. Well, that is more or less fine—I don't expect even the best philosophers to be so complete as to put forth such a magnum opus theory of existence from child psychology to technological advance. But I do expect him to at least accept the charge that simply ignoring something like a theory of meaning (which could very well fall under the scope of his method of falsification, were it not for some paradoxical results that would emerge) as "already taken care of" and yet attacking apriorism and verificationism puts him on shoddy ground, something he doesn't seem to even recognize. He attacks all positions and squeezes his own in, meaning that only his can be used to describe empirical pursuit. Which a theory of meaning would fall under, barring transcendentalism and apriorism. So is he putting forth a theory of meaning? Well, no. But has he left it any room to go? Yes, actually, he has left one spot: inside his own theory of falsification. If he hasn't left it there, then the rigor he applies to scientific theories is pretty much worthless as any theory, based on inter-subjective testing, relies on participants' communication, and can thus be no more certain than those communicative acts.

[Spiritus Mundi]

Cuba? But I don't want to go to Cuba.

Besides, I confess to neer having read Popper's work in anything but summary form.

[erislover]

How was that a hijack? The feedback system the link describes is more than just analogous to Popper's position on empiricism.

[december]

Interesting essay. ISTM that it's trying to say more than it ought to.

I actually saw the Banach-Tarski paradox presented many years ago, when I knew enough mathematics to follow it. It shows how to decompose a sphere into a small number of pieces and then put the pieces back together to form a larger sphere. Since the pieces are "non-measurable" the result doesn't contradict theory. Still, as the essay says, it does show that "mathematical axioms that seemed to be consistent with phenomenal experience could lead to dizzying contradictions with that experience."

But, so what? Many theories become silly when carried to some extreme. That doesn't mean that the theories may not be useful in more normal ranges.

Godel's result showed that a sufficiently large set of axioms must be incomplete. That is, there will be undecidable propositions. Again, so what? It's easy enough to state propositions that have no importance which may well be undecidable. E.g., The decimal expansion of pi has 10% 1's

What difference does it make if this proposition is undecidable? Such a result would have no meaning for science or real-world application, anyhow (so far as I can imagine.)

Neither of these points really detracts from the usefulness of mathematics to science.

[TVAA]

Mathematics is a language, like any other. There's nothing that mathematics can do that any other language can't. What makes mathematics unusual is that it's used with extreme precision: instead of generating a concept that we assert has some link to reality, we manipulate basic concepts and look at what we get.

[FranticMad]

the author is deconstructing the efficacy of mathematics as a tool for modeling physical reality, and claiming that it's not so miraculous after all.

The essay follows the current fad of postmodern deconstructionism which is summarized by "If you can think of it, it can't be real."

And (yawn) it includes obligatory references to Godel and chaos theory. As a math essay, I'd give it a C-. For a third year philosophy essay, I'd give it a B+. Now if it were an essay for political science or gender studies, I'm sure it would get an A+ at most universities.

[Scott Dickerson]

There are some problems with the cited essay. It's a little slippery to present Frege, for example, as giving aid and comfort to the formalists without noting that Frege was a Platonist; or in the same context to mention Russell and Whitehead and ignore their program of logicism, which I (mis?-)take to be founded on the idea that mathematics derives from the laws of traditional syllogistic (whereas Formalism incorporates additional rules of the game applicable to mathematics specificita). There are points when I'm not sure whether the author means Formalism (a distinct school of thought about how mathemtics works) or "formalism" (a general policy of reliance upon a "meta"-level system of rules). And "phenomenal science" seems an odd ideosyncracy.

But at any rate, I suppose the question before the house is: Is the evident broad and useful applicability of mathematics in empirical science due to the fact that Reality in all its parts has a form particularly amenable to mathematical description; or due to the fact that we human math-doers only regard a part of Reality as "scientifically explained" to the extent that it is mathematizable? (and add to the latter the notion that we may keep redefining what constitutes a "real" part until we DO define a part in such a way that it is mathematizable.)

From my philosophical point of view:

1) The Real is The Knowable; more significantly, we can a priori exclude from our list of Real Things any purported entities the basic definitions of which include mutually-negating descriptors (eg, "is square and is not square") or, of course, nonmeaningful pseudo-descriptors.

2) Mathematics (in ultimate idealized form) is a self-unfolding language of descriptions; where "language" explicitly refers to acts of communication.

3) It is an empirical question, a question of fact, whether our current language of mathematics, or indeed any possible language of mathematics, is able to "cover" every part of The Real. I take it Godelian Incompleteness raises a doubt. The issue for me is whether:
"some possible person"
could in principle
conceive of what it would be like
to encounter-in-knowing
some Knowable that manifested
non-mathematizability as one of its attributes.

AND, if so, whether such items are among those in which empirical science takes an interest.

I incline to the view that not all (things that we would classify as) physical events MUST belong to systems that have a complete mathematical description. To that extent and in the way I agree with the conclusion of the essay.