All Questions
Tagged with philosophy foundations
139
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What did Richard Dedekind mean exactly by his statement about generality?
But—and in this mathematics is distinguished from other sciences—these
extensions of definitions no longer allow scope for arbitrariness; on
the contrary, they follow with compelling necessity from ...
- 27
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64
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Arithmetization of Turing machines
Refer to Turing's 1936 paper, page 248, last paragraph. I present the paragraph in verbatim below :
The expression "there is a general process for determining..." has been used throughout ...
- 345
7
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1
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319
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How should one understand the "universe of sets"?
One way to understand the axioms of $\mathsf{ZFC}$ is to see them as a describing the "universe of sets" $V$, together with the "true membership relation" $\in$. The universe $V$ ...
- 20.7k
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2
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304
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On the Consistency of Non-Euclidean Geometry
I've recently found a really old "Philosophy of Math" book in my University library, and in the book it says that the it has been proven that :
if non-euclidean geometry is inconsistent, ...
- 77
2
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1
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370
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Why are we confident in the ability of ZFC to formalise mathematics if very few proofs are actually converted into ZFC?
$\mathsf{ZFC}$ is often introduced in logic textbooks as a first-order theory with equality and a single non-logical symbol $\in$. However, even stating the axioms of $\mathsf{ZFC}$ in this language ...
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Criterion of Identity for 'set'
I'm looking for different criteria of identity for the notion of 'set'. I know that the standard criterion of identity is extensionality but I was wondering if there are others. I looked around but ...
- 19
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2
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356
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How do we understand a 'universe' in the context of Mathematics?
I recently open a can of worms for myself by inquiring if there is a difference between a number as a natural or real, and got a fair answer, in doing so I came by an interesting idea about viewing ...
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604
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Do the everyday mathematician and the model theorist mean the same thing by "truth"?
In Terrence Tao's book Analysis I, the axioms of ZFC are considered to be true statements, and every other true statement in the book is proved from these axioms. Model theory is not mentioned.
This ...
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5
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342
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Is there anyway to quantify how difficult a statement is to prove in Mathematics?
Suppose we have an axiom system and theorems derived out of that axiom system, is there any way to rigorously speak about a theorem being more difficult to prove than others?
In my personal thoughts, ...
11
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4
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573
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Why is it not important what mathematical objects are?
In axiomatic set theory the term "set" and the relation "$\in$" are primitive notions. Thus, it is not defined what sets are nor what the relation is. Axiomatic set theory is ...
- 389
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Implications of $Q\vdash \lnot Con(Q)$
By Gödel's incompleteness (which holds even for $Q$ as discussed here) we have that the Robinson arithmetic $Q$ is consistent iff $Q+\lnot Con(Q)$ is consistent.
As I understand it, without further ...
- 332
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365
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Is ZFC arithmetically sound?
I apologize that this question is fairly philosophical and not purely mathematical. For the purposes of this question, I would like to take the point of view that that natural numbers are "real&...
- 4,794
4
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415
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Consistency in Hilbert's Foundations of Geometry
After reading the entry in the Stanford Encyclopedia of Philosophy on Hilbert and Frege’s correspondence regarding the former’s Foundations of Geometry, I am quite puzzled by a claim that is made by ...
- 577
4
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1
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294
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Mathematicians endorsing platonism -- examples?
Take platonism to be the view that there are abstract mathematical objects which exist independently of us as mathematicians and our language, thought, and practices.
Looking at the Stanford ...
- 55.4k
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148
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Histories in Scott-Potter set theory
I have been reading Michael Potter's Set Theory and its Philosophy. I am confused by the concept of a history, which I understand is somewhat unconventional. The definition given is as follows:
The ...
- 148