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In may 2013 I have to write a Bachelor Thesis for my bachelor Mathematics. I prefer to choose a subject which involves philosophy. At the same time I have the feeling that my university wants me to write a more technical thesis (which I do not prefer). So I want to be prepared to claim that a philosophical subject is technically enough.

What are great subjects that combine mathematics with philosophy?

I thought for example about what mathematical structures really are. What is the relation between mathematics and the reality? Are we discovering the reality or are we constructing while we learn/discover mathematics.

I have followed courses about logic: on introductionary level and some more advanced logical topics (like modal logic, many-valued logic, time logic, non-monotonic reasoning). Maybe I can combine philosophy with such topics? If anyone has a good idea, I would like to hear it!

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You might get some ideas here: mathoverflow.net/questions/94354/… –  Joel Reyes Noche Jan 5 '13 at 1:40
    
Try mathematical logic or axiomatic set theory. –  Mohan Jan 5 '13 at 1:40
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I'm not sure what exactly can a bachelor thesis be, but perhaps people in the mathematics department are expecting it to be about mathematics neto. What you probably want to do could perhaps better fit in graduate school, unless you made heavily philosophical flavoured undergraduate title...\ –  DonAntonio Jan 5 '13 at 1:41
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Ask your advisor. –  Potato Jan 5 '13 at 1:48
    
I don't know if it counts as related to philosophy, but I always enjoyed mathematical, sometimes counterintuitive paradoxes such as Monty Hall's problem. More related to set theory, you have e.g. Russell's paradox and the Banach-Tarski paradox. –  TMM Jan 7 '13 at 20:23

3 Answers 3

up vote 1 down vote accepted

You may be interested in this paper by Barry Mazur at Harvard.

"When is one thing equal to some other thing"

What motivated me to mention it to you is a remark early on:

"I don’t even see how questions about these issues can even be raised within the framework of vocabulary that people employ when they talk about the foundations of mathematics for so much of the literature on philosophy of mathematics continues to keep to certain staples: formal systems, consistency, undecidability, provability and unprovability, and rigor in its various manifestations."

So I thought that perhaps this might open a somewhat different door.

It is linked in the "Further Reading" section of the first answer to this very interesting question here:

Why do we look at morphisms?

There are two sections that might be of special interest:

-- Defining Natural Numbers

-- Equality vs. Isomorphism

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This looks good. Thank you for sharing. I will definitely consider it. –  MSKfdaswplwq Jan 24 '13 at 11:30
    
My pleasure. Best of luck –  Andrew Jan 24 '13 at 11:39

It very much depends what maths you have already done. It would be a lot of work to both learn enough about a new topic $X$ and then engage with broadly philosophical discussions about $X$.

But if you have done/are doing a course in set theory, there are obviously interesting questions to be discussed there. You are unlikely to be in a position to write usefully about e.g. large cardinal axioms ("do we need new axioms?"), but there are more basic questions. What's so great about ZFC? Why this set theory rather than one of its rivals? Does it even best encapsulate the hierarchical conception of the set universe? Michael Potter's Set Theory and its Philosophy could be useful here.

If you have done/are doing a course on computability, there are other interesting questions to be discussed. The significance of limitative results like Gödel's theorem and/or the unsolvability of the Halting Problem. Or: why should we believe the Church-Turing Thesis? My Introduction to Gödel's Theorems could be useful here.

Topics in these areas would give you a chance to show some mathematical knowledge (presumably a requirement). Very general discussion about the notion of structure is unlikely to go down so well for a maths degree (though you can only find out by consulting your advisors), while discussing e.g. the light thrown on on the notion of structure by category theory will require going too far beyond the BA level maths you know.

So my summary advice: for a BA-level thesis, look for some relatively narrow topic related to sufficiently advanced topics you already know something of. Avoid over-general topics like "what mathematical structures really are". [I'd say the same, mutatis mutandis, for BA level philosophy theses!]

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TY. This is a useful answer. In addition to my question: I have followed courses about logic: on introductionary level and some more advanced logical topics (like modal logic, many-valued logic, time logic, non-monotonic reasoning). Maybe I can combine philosophy with such topics? If anyone has a good idea, I would like to hear it! –  MSKfdaswplwq Jan 7 '13 at 19:29
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@JoyeuseSaintValentin: If you want to investigate questions like "what is the relation between reality and mathematics?" and "what are mathematical structures?" and enjoy a logical approach, can I suggest looking into some of the ultrafinitists and related? Some take the approach that mathematics is a physical process, something occuring in a universe with limited finite resources, and so one cannot validly reason about things with infinite characterisations. There are several flavors, and plenty of opportunity to discuss semantics, models, and what mathematical reasoning is about. –  ex0du5 Jan 7 '13 at 20:18
    
What are ultrafinits? –  MSKfdaswplwq Jan 11 '13 at 11:42

I guess you could try to look something about foundation of mathematics, which I believe it's considered a subfield of the broader philosophy of mathematics. There's a lot of interesting stuff here set theory and logical system which I think are interesting from a philosophical point of view, although they are really mathematical and involve also a lot application in ,maths but also in computer science (in particular I'm aware of the interconnection of this subject with calculability theory).

Hope this help.

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