Talk:Philosophy of mathematics/Archive 2

heh. this is going to be *so* weird to people who think math is real...

It's also a total mess as an encyclopedia article.

This article should describe the various different philosophies of mathematics.

it describes three extreme points of view, those being that it is fundamental physical and unchangeable, that it arises entirely from the proof process followed by mathematicians, and that it arises entirely from the structure of the human mind. A fourth extreme point of view, that it is entirely a colonizing process, I didn't deal with, as it's so postmodern.

Also, I doubt it should be "weird" to anyone. Anyone who works with math pretty soon hits some philosophical issues.

yeah but they breeze past them - of 150 mathematicians in one graduating class I know, only two of them paid any attention to the field proper, and they disagree sharply, falling one each into the "embodied mind" and "part of the physical universe" camps. This is not an easy subject to really explain.

A lot of scientists (particularly if you get into the social sciences) don't consider mathematics neutral.

It differs from the philosophy of science by not taking mathematics as a neutral point of view - rather, investigating such subjects as the willingness to accept mathematical proofs, the validity of induction or analogy, what combination of metaphors constitutes an isomorphism, mathematicians' social capital and the meaning of the well-known collaboration graph.

A lot of 'hard scientists' don't consider 'social science' to be science -- they point to the lack of falsifiability in many of their theories, and assert that abandoning objective standards of mathematical and logical proof removes the basis for falsifiability, leaving only opinion, fashion, and popularity contests.

There are physicists who do think that. But that's a relatively small minority opinion (something like 10% or so). My experience with "math-fetishists" is that they actually tend to be social scientists. Very few physicists (or even mathematicians) seem to believe that an idea that can't be expressed in mathematics isn't an idea at all, however I've known/read a number of social scientists that seem to think that. The problem is that regarding mathematics in social science *greatly* limits the hypothesis that you can form and the theories that you can test. Also, pretty much anyone who does qualitative research would disagree with non-mathematical means non-falsifible, and I'm pretty sure that most physicists don't think this (even though I know of a few who do).

The problem is that regarding mathematics in social science *greatly* limits the hypothesis that you can form and the theories that you can test.

That's probably the attraction for us math-fetishists. Science must be so easy for the math-challenged: you simply have to think of things, write papers and win the support of your peers, without any real chance of bumping up against nasty cold non-human reality.

The trouble is that just because something is stated in terms of mathematics doesn't automatically make it more objective, precise, testable, falsibility or correct. I get *REALLY* annoyed at people who think it does. (I should point out that I have a Ph.D. in physics.)

all of this is what leads to "reasonable method" ideas. Science itself, hard or soft, is basically in doubt at the moment... neutrality of its process is questioned for a lot of reasons... Falsifiability as a doctrine has little or no applicability to basic mathematical axioms... which are too abstract to falsify....

This article is still a mess. It's confusing, misleading, and in some places, downright wrong....

Wrong? Where? Your strange misreading seems to be the problem here:

I guess I'll start by removing the most egregious errors

asks what makes one theory more acceptable than another despite imperfect empirical validation and limits of the scientific method - and assumes mathematics as a neutral point of view.

I'm not sure what neutral point of view means in this context. If the statement is that mathematics doesn't influence how scientists view the world, then its false.

other way around - are you actually *reading* this or just reacting to it? it is quite true that scientists assume mathematics as a neutral point of view, and that philosophy of science has no basic critique of mathematics itself, except for some of the social sciences as above.

This is gibberish. It's also inaccurate

and collaborators, actually mapped onto the physicists' particle physics foundation ontology or if they were simply another sacred geometry like that of Plato - a useful but limited model that awaited understanding of some deeper ontology.

Explain then why various commentators refer to "Platonic neorealism" in both physics and in mathematics, both arising seemingly from overbelief in Euler's Identity...

So did Erdös really do work on foundations? Or the 'fundamentals' of number theory? It's not a field I associate with his name. Anyone? Matthew Woodcraft

Agreed. Erdos' program was to re-prove many basic proofs in number theory without any reliance on complex analysis whatsoever... that in itself changed the fundamentals, as it removed reliance on a whole set of methods and assumptions. The well known graph of Erdos' collaborators drives new work in graph theory and research into collaboration on research. "Paul Erdos, more than anyone else, did the most to make mathematics into a social activity"

I see. Did he do any work on foundations? In Erdoös's lifetime, number theory wasn't at the foundation of mathematics; it was typically built up from set theory, yes? Matthew Woodcraft

Various people who understand nothing of the modern theories are hacking this. No one says that scientists aren't influenced by mathematics, quite the other way around, they believe in it overmuch, and in "falsifiability" overmuch, and that ideology is clearly guiding the commentary in the above.

Falsifiability is ideology. Other than strict followers of Popper, it's hard to find anyone who believes that cognitive bias and infrastructural prior investment don't play huge roles in determining what constitutes a "disproof".

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Why is "metaphysics" relevant, and "ethics" not? Philosophy of mathematics is often framed in more theological terms of ontology, morals, and cosmology... which is not the ethics/epistemology/metaphysics distinction used in philosophy. In particular Godel neatly moved from one to the other, and Wittgenstein and Russell... most of the major figures. Erdos has some very strange compilations of conventional cosmology, which are called Erdosisms.

That's certainly not the usual referent of the term 'Erdösism'. And I don't believe that either morals or cosmology 'often' come up in discussion of the philosophy of mathematics.'

Perhaps it is more correct to say there is a "philosophy of science" but a "theology of mathematics"...?

Snipped from article - the quote preceding them suffices, and these just add blur:

In terms of which, the reasonable method is the best description we can make of the best human mind, and the foundation ontology is the best description we can make of "reality, something out there to be discovered."

As use of mathematics is not confined to scientists making predictions, but is employed also in law and economics and political science, most researchers find it appropriate that "we" be a somewhat larger (some say less disciplined) group than physical scientists, and that reasonable method be perhaps more inclusive than feasible method or scientific method.