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Comment by mablap | Hacker News Reader

    Analytic philosophy has made my mind a razor. [...]

Formal mathematical training (graduate) will do that to you too. I often find myself thinking with abstract algebra ideas (groups, vectors, vector spaces, commutation relations, etc).

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I studied both. They're great.

I'm well trained in mathematics. It's certainly healthy. However, I often find that outside of their professional work, mathematicians are more willing to part with rigueur than the philosophers across the hall. Notice how most mathematicians aren't particularly interested in foundations of mathematics, for example.

I'm not sure if that's really true about mathematicians lacking rigour.

I do recall a logic professor of mine jokingly say that mathematicians were afraid of logicians, and (more seriously) that the former thought the latter were unnecessarily precise. (I hope I'm remembering this right and not misrepresenting what he said)

I'm not sure about the superiority in rigour in the rest of philosophy (even if we narrow the meaning of that term to just analytic philosophy, some of whom sure are fond of their logic, symbols and attempting to sound "scientific" or rigorous). If anything, I'd say those analytic philosophers have math (or hard science) envy.

You're right about mathematicians not being very interested in the foundations of math. That's a consequence of the failure of the great foundational project in the early 20th century. From what I understand, most mathematicians have decided that such a foundation is not possible, and view themselves as moving on to doing the very practical business of math anyway. After all, the lack of a foundation has not prevented them from achieving many interesting (and often useful) results.

That said, there has continued to be interest in foundations for math from some logicians, and (at least according to my logic professor) it still might be possible to found math on logic yet. That's quite a bit out of my league, however, so I'm afraid I can't elaborate much. But if you're interested in that, I do recommend taking some courses in symbolic logic and in the philosophy of math.

I can only assume his persona and commentary is an elaborate satirical parody of the snooty narcissistic philosopher (but you never know.. Poe’s Law and all that). To quote from saintzozo’s website:

Mathematics is easy.

If you think it's hard, you are retarded.

proof

Any true mathematical statement is logically equivalent to the axiomatic framework within which it occurs. If you do not understand such a statement, there are only two possibilities:

1. You do not understand the axioms.

Axioms are chosen so that they are evident a priori (eg. the probability of all disjoint events must sum to unity). If all the axioms are not clear to you, there is something seriously wrong with your reasoning faculties.

2. You do not understand logic.

If you understand the axioms, then the only thing that could prevent you from understanding a true mathematical statement is an inability to reason logically. Such a crippling deficiency defines what it means to be retarded.

This case analysis exhaustively proves that if you don't understand math, you are retarded. ■

corollary

Now one must be wary of students formally enrolled in programs of study devoted to mathematics (and its bastard child, computer science).

These people misunderstand mathematics (read: are retarded) to such an extent that they have resorted to paying other people money for instruction in the obvious. [...]

Sir, I own the copyright to this material and I would appreciate if you do not reproduce my work without permission. For some reason, this site no longer appears to function with my original account.

Axioms are chosen so that they are evident a priori

Wonder why it took 2000 years to came up with alternatives to the 5th postulate, if all three variants (Euclidean, elliptic, hyperbolic) is so evident apriori?

I wouldn't say there is maths envy in analytical Phil.

Stuff like modal logic , ontology and descriptive logics , a large variety of para-logical systems and epistemically logical systems, etc are all philosophy proper.

Lots of professors have a background in maths, even continentals (e.g Husserl).

I think mathematical rigour is the holy grail but the objects of the philosophical world are often not easy to coerce into such a form hence the piecemeal visage.

I don't know about willingness. There's one way to back up your point though. Philosophers learn to apply rigor in very broad situations, getting used to usisng it everywhere.

Mathematicians apply it very rigorously, but mostly in math. Applying that to other areas without specific training is difficult. For this reason, academic philosophy makes a person more rigorous in life generally.

Until you meet a math problem, of course.

Not that learning anything as rigorous and precise as math won't teach you clearer thinking - of course it will.

I'm a mathematician working in industry and I'm not at all interested in foundations. That's partly because I trust that all that stuff was worked out in the last century, and partly because there's so much more exciting stuff in math to be interested in instead.

Your belief that foundations were worked out in the last century is completely mistaken. Very serious and concerted efforts at foundations were made then, but they famously and dramatically failed when apparently unreconcilable contradictions were found at the core of these foundations. Then the foundational effort was largely abandoned and mathematicians moved on.

I suppose back around that time and up to a certain time after this happened, most mathematicians were aware of this history, but perhaps they aren't any more, if yours is a representative view.

So then it comes down to interest.

But many mathematicians were interested in foundations around the early part of the 20th Century. Now they're generally not.

This shift makes me wonder how much of mathematical foundations are actually "inherently interesting" (if there is such a thing) to mathematicians and how much has to do with fashion, and if it foundational projects among mathematicians will ever become fashionable (or "interesting") again.

Can you name three unreconcilable contradictions in ZFC?

"The most comprehensive formal systems yet set up are, on the one hand, the system of Principia Mathematica and, on the other, the axiom system for set theory of Zermelo-Fraenkel (later extended by J. v. Neumann). These two systems are so extensive that all methods of proof used in mathematics today have been formalized in them, i.e. reduced to a few axioms and rules of inference. It may therefore be surmised that these axioms and rules of inference are also sufficient to decide all mathematical questions which can in any way at all be expressed formally in the systems concerned. It is shown below that this is not the case, and that in both the systems mentioned there are in fact relatively simple problems in the theory of ordinary whole numbers which cannot be decided from the axioms."

From "On Formally Undecidable Propositions of Principia Mathematica And Related Systems" by Kurt Gödel, 1930

https://en.wikipedia.org/wiki/On_Formally_Undecidable_Propos...

http://www.csee.wvu.edu/~xinl/library/papers/math/Godel.pdf

I am aware of Godel, thanks, and of course that choice / Zorn's / well ordering (not to mention CH) are independent of ZF (and in particular that's why I mentioned ZFC). I'm also willing to accept the existence of an inaccessible cardinal, which bears on ZFC's consistency.

Ultimately my belief is this: there was a very real crisis in foundations at the end of the 19th century and over the next several decades this was fixed as best as it could be. The edges of the foundation are not perfect, the edges of the foundation cannot be perfect, but the edges of the foundation have been pushed back so far that for nearly every working mathematician they're good enough. (and if the algebraic geometers need Grothendieck universes, I'm ok with that)

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