Philosophy of Math: Interesting Leads & Ideas

[Abscissas]

Dear everyone,

So I am looking into the broad topic "Philosophy of mathematics." Do you guys have any interesting leads that relate to the topic I can follow? I have no particular direction I want to go in, I am just looking for cool stuff, ideas, and people you think are interesting with regards to the topic.

-x

 

[jedishrfu]

I would look to YouTube channels numberphile and 3blue1brown for interesting areas of math.

The domain of science channel has an excellent short called the map of mathematics which is just phenomenal.

 

[Abscissas]

I would look to YouTube channels numberphile and 3blue1brown for interesting areas of math.

The domain of science channel has an excellent short called the map of mathematics which is just phenomenal.

I just checked out the map of mathematics, loved it. I think what I am looking for is when they mentioned the foundations of mathematics. I am imagining that I want to know the thought processes or typical mind sets mathematicians had during a particular era. What our interpretation of math is, and what math it self is.

Im going to check out numberphile and blue brown and I'll let you know if I find anything particularly cool.

-x

 

[Abscissas]

Just to keep you guys updated/ I am going to use this as a research journal,

I havn't gone through numbephile or blue brown just yet, my apologies
I wikipedia'd philosophy of math and found some cool looking terms
I went to my school's data and found a couple of cool looking papers:
- Interpreting the Infinitesimal Mathematics of Leibniz and Euler
-Michelangelo's Stone: An argument against Platonism in Mathematics
- Poincare on the Foundations of Arithmetic and Geometry
- Movement, memory, and mathematics, Henri Bergson and the Ontology of Learning
-Some Obstacles Facing a Semantic Foundation for Constructive Mathematics
-Depth -- A gaussian tradition in mathematics
-Matheme and Mathematics: On the main concepts of the philosophy of Alain Badiou
-Mathematical work of Francisco Varela
-Wittgenstein on pure and applied mathematics

These are a lot to read through, but as I finish them I'll let you guys know what I find in there that is kind of cool. I'll be watching numberphile and blue brown when I get bored of reading.

-x

 

[fresh_42]

I am imagining that I want to know the thought processes or typical mind sets mathematicians had during a particular era.

I'd try to read everything about Hilbert's program. It condenses the nature as well as the limits of mathematical knowledge. It's results made mathematics more similar to physics than single theorems ever could have done, as it shows, that there are indeed limits. E.g. to use the axiom of choice is in the end a deliberately made decision to achieve some results, although it can't be based on anything else. That's a bit like how physicists use mathematics: If it fits my experiments, it works.

 

[nomadreid]

Hilbert's program. It condenses the nature as well as the limits of mathematical knowledge. It's results made mathematics more similar to physics than single theorems ever could have done, as it shows, that there are indeed limits.

More precisely, it was the failure of Hilbert's program that set the strongest limits of mathematical knowledge. Gödel's two incompleteness theorems, with various refinements later, to be even more precise. But one shouldn't compare the limits on mathematics (which are epistemological limits) with the limits in physics (which, if one is referring to the limits brought about by quantum theory, are ontological limits) too closely. otherwise put, undecidability is not the same as indeterminacy.

 

[fresh_42]

... otherwise put, undecidability is not the same as indeterminacy.

I'd say indeterminacy isn't a cause of limits, rather a cause of language, if at all. The physical limits as I see them are more of a practicable nature. E.g. we cannot always provide as much energy as we would like to. Of course, this is still different from undecidability and the comparison lacks correspondence. I only wanted to point out, that although some mathematical tools cannot be based on a natural view, mathematicians had to learn to get along with this fact. One could have expected a split as in many other areas, but this wasn't the case. Inconvenience was just separated into another, not very famous branch of mathematics, or even better: defined out of the way, as the set of all sets containing itself as element. To this respect, modern mathematics is similar to physics, in so far as unpleasant facts are simply disregarded as long as consistency can be achieved. From a pure philosophical point of view, there are of course fundamental differences in this comparison. However, the attitude of convenience is similar: mathematicians in their daily business don't really care, that the AC is outside of arithmetics, and physicists renormalize that mathematicians would run away.

 

[nomadreid]

Ah, good that this was explicitly clarified, so that the OP will be aware that words like indeterminacy, limits, etc. have more than one meaning.

 

[jedishrfu]

My suggestion is to read some books like Bell’s Men of Mathematics which may be a bit dated, and Jan Gullberg’s book on history of math and maybe even Math 1001 by Elwes. They are pop culture books but do get into math stories. Elwes book cover more of the current open problems in math over many areas.

There’s also the Princeton Math Companion for heavier reading in math.

https://www.amazon.com/Mathematics-Birth-Numbers-Jan-Gullberg/dp/039304002X

https://www.amazon.com/Maths-1001-Dr-Richard-Elwes/dp/1554077192

https://www.amazon.com/Men-Mathematics-Touchstone-Book-Bell/dp/0671628186

https://www.amazon.com/Princeton-Companion-Mathematics-Timothy-Gowers/dp/0691118809

 

[SSequence]

Mention of another problem. It is whether there are examples of subsets (of N) that correspond to solution of Post's problem, that describe a natural problem. Post's problem is simple to state in terms of computational/turing reducibility. Its solution (I haven't studied it) however wasn't simple and used clever tricks**.

Originally there were two (potential) expectations:
(1) The resulting solution would in some way be useful in field of algorithms (that tends to be concerned with problems of more practical nature I guess). However, as far as I know, not much was found in this direction.

(2) Natural examples (or perhaps examples linked with some other math fields?) would be found whose solution could be described by the solution of the problem. I don't think this materialized either.

Anyway, some people criticize the study of resulting field (significant part of classical computation theory) because of (2). I don't quite agree with it of course. Anyway...

** When Paul Cohen (according to himself I think) first saw the solution he was disappointed because he wanted to do something involving clever tricks like that himself. So it seemed to him that he missed an opportunity.
This is paraphrasing of course.

 

[osilmag]

This is a video of the Top Ten Math Equations that Changed the World.