Avigad’s Does philosophy still need mathematics and vice versa

Jeremy Avigad Does-philosophy-still-need-mathematics-and-vice-versa Aeon

This is an exceptionally quick comment, even by my standards. Sorry about that. The article provides a respectable contemporary philosopher’s view, with which I contrast my own: particularly where I find the philosopher’s view troubling and possibly dangerous.

While I agree with the conclusion, much of the discussion of ‘the problem’ fails to recognize that mathematics has moved on in the last 100 years or so.

According to Descartes, philosophy is a matter of discovering general truths by finding properties that are shared by disparate objects, in order to understand the features that they have in common. This requires comparing the degrees to which the properties occur. A property that admits degrees is, by definition, a magnitude. …

Plato held mathematics in great esteem, and argued that, in an ideal state, all citizens, from the guardians to the philosopher kings, would be trained in arithmetic and geometry. In The Republic, his protagonist Socrates maintains that mathematics ‘has a very great and elevating effect’, and that its abstractions ‘draw the mind towards truth, and create the spirit of philosophy’.

I tend to agree with Socrates, but mathematics has moved on since Plato and Descartes. As here:

In the 1930s, the Austrian logician Kurt Gödel proved important results known as the incompleteness theorems, which identify inherent limits to the ability of the axiomatic method to settle all mathematical truths. Via the mathematical modelling of mathematical practice itself, logic therefore gave us a clear account of the nature and extent of mathematical reasoning.

Logic may ‘give’ ‘us’ a clear view, but I doubt that many have ‘taken it’. (I see this as a significant problem, not just an academic quibble. Hence my blog.)

Both the positive results and the negative, limitative results were valuable: having a clear understanding of what a particular methodological approach can and cannot be expected to achieve [ideally] serves to focus enquiry and suggest new avenues for research.

But when, in practice, is this ideal achieved, even approximately?

Insofar as it is possible to provide compelling justification for doing mathematics the way we do, it will not come from making general pronouncements but, rather, undertaking a careful study of the goals and methods of the subject and exploring the extent to which the methods are suited to the goals

I’m not sure how to read this. What is he thinking about justifying? What does he think that mathematics ‘is’ and how does he think we ‘do’ mathematics? (Hopefully, not what Descartes thought.) Who are we trying to justify mathematics to?

By the simplest reading any justification of mathematics would seem to require some justification of its application, which necessarily goes beyond pure logic. The best one could do to justify mathematics  would seem to be Keynes’ argument: it seems to work, doesn’t it? (Imagine the 20th century without it.) On the other hand what seems to me more in need of justification is not the way we justify how we ‘do’ mathematics as such, but the way we apply it.  We need critique, not ‘justification’.

The field’s narrow focus on logic suggests another explanation for its decline. Given that the philosophy of mathematics has been closely aligned with logic for the past century or so, one would expect the fortunes of the two subjects to rise and fall in tandem. Over that period, logic has grown into a bona-fide branch of mathematics in its own right, and in 1966 Paul Cohen won a Fields Medal, the most prestigious prize in mathematics, for solving two longstanding open problems in set theory. But there hasn’t been another Fields Medal in logic since, and although the subject enjoys some interactions with other branches of mathematics, it has not found its way into the mathematical mainstream.

This is an insightful observation. But ‘the philosophy of mathematics’ is quite different from mathematics as such. Surely the question is why have logic and mathematics been so independent?

(1) Logic-based methods have yet to yield substantial success in automating mathematical practice, whereas (2) statistical methods of drawing conclusions, especially those adapted to the analysis of extremely large data sets, are highly prized in industry and finance.

Worrying. (1) What about Turing? (2) How is being ‘highly prize’ relevant here?

Computational approaches to linguistics once involved mapping out the grammatical structure of language and then designing algorithms to parse down utterances to their logical form.

Okay, but what kind of logic are we talking about here? Are they ‘pragmatic’ logics, applied ‘pragmatically’ or ‘logical logics’ applied logically?

Whereas mathematics seeks precise and certain answers, obtaining them in real life is often intractable or outright impossible.

Yes, but don’t we all ‘seek’ certainty and fail to find it? (And doesn’t logic explain why?)

In such circumstances, what we really want are algorithms that return reasonable approximations to the right answers in an efficient and reliable manner.

We may ‘really want’ algorithms, but are they appropriate?

Real-world models also tend to rely on assumptions that are inherently uncertain and imprecise, and our software needs to handle such uncertainty and imprecision in robust ways.

Yes please!

Evidence for a scientific theory is rarely definitive but, rather, supports the hypotheses to varying degrees.

This might seem harmless, but earlier the article states “A property that admits degrees is, by definition, a magnitude.” I would suggest some Whitehead, Keynes, Russell or Turing as an antidote.

If the appropriate scientific models in these domains require soft approaches rather than crisp mathematical descriptions, philosophy should take heed. We need to consider the possibility that, in the new millennium, the mathematical method is no longer fundamental to philosophy.

Okay, but maybe also consider the possibility that we just need to select and apply mathematics differently.

Realistically, deep learning is only part of the larger challenge of building intelligent machines. Such techniques lack ways of representing causal relationships (such as between diseases and their symptoms), and are likely to face challenges in acquiring abstract ideas like ‘sibling’ or ‘identical to’.

Surely we need to understand ‘causality’ before we can represent it?

The question, then, is not whether the acquisition of knowledge is inherently hard or soft but, rather, where each sort of knowledge is appropriate, and how the two approaches can be combined.

Agreed.

Mathematics has been remarkably resilient when it comes to adapting to the needs of the sciences and meeting the conceptual challenges that they generate. The world is uncertain, but mathematics gives us the theory of probability and statistics to cope.

Probability is one type of uncertainty, but what about others? (See my blog.)

Natural and designed artifacts can involve complex networks of interactions, but combinatorial methods in mathematics provide means of analysing and understanding their behaviour.

This seems like a very strong scientific statement. How could it be justified?

But to make sense of the world, what we really need is the serenity to accept the things we cannot understand, courage to analyse the things we can, and wisdom to know the difference.

Is this saying more than Russell et al?

But the philosophy of mathematics has served us well in the past, and can do so again. We should therefore pin our hopes on the next generation of philosophers, some of whom have begun to find their way back to the questions that really matter, experimenting with new methods of analysis and paying closer attention to mathematical practice. The subject still stands a chance, as long as we remember the reasons we care so much about it.

Okay, but possibly we have overlooked some old philosophers?

Dave Marsay

 

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