4
votes
4answers
380 views

The standard role of intuitive numbers in the foundations of mathematics

In my career I've been formed mostly in the formal side of mathematics, that is, standard set theory and every classical branch of mathematics that uses set theory. However, I am not quite sure about ...
1
vote
1answer
48 views

What exactly is a property?

How is a property $P$ formally defined in mathematics? I mean for example if $f$ is a morphism from an object $X$ to $Y$ in some category, then somehow I feel that "has codomain $Y$" is too broad to ...
5
votes
2answers
79 views

What is gained by internalizing LST (the language of set theory)?

I'm reading up on Gödels constructible universe L in the book "Constructibility" by Devlin, and by comparing his text with texts like Kunen and Jech, there is one thing in particular that he's doing ...
-4
votes
1answer
91 views

Is there an “end”? [closed]

This question may seem silly, but I nevertheless think that it is worth wondering over: is mathematics itself finite? As I understand, mathematics is a study of form and existence under constraints, ...
1
vote
1answer
79 views

A Simpler Characterization of Inductive Definitions?

While reading appendix A of John Harrison's "Handbook of Practical Logic and Automated Reasoning" a somewhat advanced theorem is appealed to as a prerequisite for characterizing when an inductive ...
8
votes
2answers
884 views

Definition of “non-constructive proof”

I was wondering if it is possible to define exactly what a non-constructive (nc) proof is. I have often seen the concept associated with the use of principles such as the axiom of choice or the law of ...
2
votes
1answer
222 views

Truth and undecidability

I believe this is more of a philosophical question. Given a consistent theory T and a statement S independent of T. Can S be true or false in T? (I don't see any contradiction with that) I read that ...
25
votes
3answers
931 views

Rejecting infinity

I've heard about mathematicians who defend a strictly finite conception of mathematics, with no room for infinity. I wonder, how is it possible for these people to do this? Are there any concepts that ...
11
votes
1answer
347 views

Formalizing metamathematics

I am reading historical/philosophical stuff on the concept of "metamathematics" and am by now quite confused. Several questions emerged, but they are probably somehow confused and interrelated, I ...
15
votes
5answers
680 views

Definition of definition

I was wondering if there is a good way to "define" what definition means exactly in mathematics. Since the answers may be subjective or philosophical, I want to ask only for references on this topic. ...
8
votes
5answers
1k views

Common misconceptions about math

YARFMO (Yet another reposting from Mathoverflow) ;-) The more you know about math the more you find conceptions previously thought correct to be false: 1.) math is not as exact as many believe - in ...
1
vote
1answer
190 views

Making meaning of mathematical “bridges”

I apologize for posting such an untechnical question, but with responses it could surely be posed in a better form. I'm a math noob, but I've seen (as we all have) a few examples of "connections" ...