Questions tagged [philosophy-of-mathematics]

Philosophy of mathematics asks questions about mathematical theories and practices. It can include questions about the nature or reality of numbers, the ground and limits of formal systems and the nature of the different mathematical disciplines.

Filter by
Sorted by
Tagged with
2
votes
1answer
257 views

Is there a formal name for a false 'co-contradiction'?

We can formally write a contradiction as, p and (not p); these statements in classical logic will always evaluate to false. Now, a true contradiction means that actually this evaluates to true. Now, ...
3
votes
2answers
310 views

Will fundamental physics become an art? (That is driven solely by aesthetics)

Famously Einstein said if he had to choose between a beautiful theory and an ugly fact he'd take the theory every-time. Dirac remarked that he always followed beauty. It seems to me that aesthetic ...
1
vote
2answers
173 views

How does Lowenheim-Skolem theorem prove the relativity of mathematical models?

Stuart Shapiro mentioned in his book Thinking about Mathematics that Lowenheim-Skolem theorem showed the "relativity" of a model in mathematics. What does it actually mean? What does the phrase "to ...
5
votes
2answers
613 views

Is Gödel's incompleteness theorem still valid if one uses a higher-order logic?

Gödel's incompleteness theorem is wholly formal (in my understanding), and relies on a proof system that I assume is first-order. Does it make any difference to the theorem if higher-order logic is ...
0
votes
3answers
231 views

Can we claim maths is finite?

By Godel's incompletness theorems we know there are problems that cannot be proven. Is this sufficient to claim mathematics (the set of axioms and theorems) is finite? As a counter-argument, we might ...
3
votes
2answers
245 views

What is the proper role of foundations in rigorous mathematics? [Paused] [closed]

[This question needs a major rewrite. Please regard it as 'paused' and inactive until I find the time to clarify the issues of concern. Thank you.] Suppose we want to write very rigorous mathematics. ...
3
votes
1answer
299 views

Does the structural interpretation of addition add something new to the philosophy of number?

Addition is a very basic mathematical operation. It seems that there is nothing more that can be really said about it. As an idea it has been around for millenia. Of course originally it was solely ...
3
votes
2answers
134 views

Do knowing quantities, which are measurable imply that one knows numbers?

Does a kid, which learns the meaning of the term "distance" (or any other expressions which might be thought of as physical quantities) automatically also develope a concept of numbers? If I know ...
3
votes
1answer
390 views

Understanding fundamental questions of life via mathematics

I am a university student of mathematics and is also interested in philosophy, mainly spiritualism. Besides knowing that mathematics is, in general, also a study of how the world works, I do not know ...
4
votes
1answer
691 views

What is the impact of paraconsistency on Gödel's theorem?

Russell's paradox forced a restriction of the natural abstraction principle (that every predicate determines a set) so that set theory could be consistent; the standard one being ZF. However ...
8
votes
1answer
554 views

In Gödels Incompleteness theorem what is the notion of truth?

The entry on Gödels Incompletenss theorem in Wikipedia says: Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for ...
0
votes
6answers
791 views

Possibly, 7 isn't prime [closed]

I'm curious if in the philosophy of mathematics (or perhaps the philosophy of modality), the following has been proposed: There exists something like an imaginary (but not complex) number i such ...
1
vote
1answer
161 views

Mathematics Essential

I started reading History of Philosophy and readily noticed that the origins of our actual natural sciences were due to the proper use of inductive logic. Our Physics/Chemistry and Biology all are ...
4
votes
1answer
267 views

What happens when we drop the condition that proofs are finite strings of inferences in Gödel's incompleteness theorem?

Gödel's incompleteness theorem shows that there are sentences that are undecideable, that is they nor their negation can be proved. This theorem operates purely syntactically or formally, that it ...
1
vote
3answers
226 views

Is full semantics in higher order logic philosophically justified?

2-logics quantify over predicates, 3-logics over predicates of predicates and so on. Unlike 1-logic where we have nice meta-logical properties of soundness, completeness & effectiveness - these ...
1
vote
1answer
144 views

What useful results or proofs use the higher order expressiveness in higher logic?

According to this article Quine has criticized higher-order logic (with standard semantics) as "set theory in sheep's clothing". Quine's criticism focuses on the lack of an effective, sound, ...
1
vote
1answer
442 views

Is model theory (for logic) a kind of type theory?

Model theory is applied to axiomatic systems to give them an interpretation. Now consider a logic in axiomatic form. When we consider a model of this logic and some sentence in the logic, each ...
1
vote
1answer
190 views

Does transfinite induction indicates limitations of Agrippa’s Trilemma?

Michael Dorfman stressed the following unavoidability in many answers: Note that due to Agrippa's Trilemma, there are only three things logic could possibly be founded upon: unsupported axioms we ...
6
votes
2answers
397 views

Criticisms of Categorical Foundations In Logic

One of the modern trend amongst mathematical philosophers has been the application of categorical foundations to logic. Lawvere is probably the best known for his use of closed cartesian categories in ...
2
votes
2answers
107 views

Is application of the frequentist interpretation of probability rigorously invalid in the real world?

The frequentist interpretation of probability states that the probability of a currently unrealized event occurring is the limit of its relative (usually temporal) frequency in a large number of ...
4
votes
2answers
400 views

How do correspondence theories handle statements like these?

I have a number of true statements. Each of these statements is a case where I have difficulty seeing how (assuming physicalism) the statement could correspond to a state of affairs. My question is: ...
5
votes
1answer
353 views

What does “aggregative mechanical thought” mean in Frege's works?

In *The Foundations of Arithmetic: A Logico-Mathematical Enquiry Into the Concept of Number" by G. Frege pages XV and XVi we read: A typical crudity confronts me, when I find calculation ...
7
votes
4answers
850 views

Is there any connection between Structuralism and Category Theory?

Having only the a very cursory knowledge of Structuralism, there does appear to be some points of coincidence: Structuralism: Individual elements of culture must be placed within a System/Structure. ...
5
votes
3answers
421 views

Is discreteness an emergent property?

The Riemann zeta function is a continuous function which encodes the properties of the primes; string theory, a proposed theory of particles, considers continuous objects; through QM discreetness of ...
2
votes
1answer
590 views

Can sheaf-theory help interpret Quantum Mechanics?

The Copenhagen interpretation posits a boundary in the World between the observer and the non-observer (that is the rest of the World). There is knowledge (Observables measured) associated with each ...
5
votes
1answer
570 views

Has there been any success in using modal logic to interpret Quantum Mechanics?

Quantum Mechanics rather famously has problems in interpretation - straightforward realism doesn't appear to work. Is there any work with modal logic that throws light on this question? The SEP has ...
5
votes
2answers
217 views

Is the mathematical notion of a “standard model” a metaphysical or a (purely) epistemic distinction?

When doing mathematics and providing models that satisfy a given theory, we differentiate between standard and non-standard models. Now, assume you are a platonist and believe that the objects ...
5
votes
2answers
207 views

Can Arithmetic recreate the transfinite hierarchy of Set Theory?

Can arithmatic when codified by the first-order Peano Axioms recreate the transfinite (cardinal) hierarchy of Set Theory (ZFC)? I suspect not, simply because we have no formal means of creating a set ...
4
votes
1answer
710 views

Was Kants formulation of mathematics as synthetic a priori a forerunner to the Russellian campaign to reduce mathematics to logic?

Kant showed that mathematics was synthetic a priori. For example the laws of arithmetic or of euclidean geometry, and noted that this had escaped the notice of previous thinkers, they had assumed them ...
0
votes
3answers
1k views

Do the Existence of Mobius Strips Prove that Mathematical Platonism is True?

Consider Mobius strips (eg. strips of paper with one or more half-twists joined together at the ends--for example, a Mobius strip with one half-twist has the interesting property of having only one ...
0
votes
1answer
493 views

Implication rules problem

P -> Q is equivalent to ~P v Q, so why isn't P -> ~Q equivalent to ~P v ~Q? I can't figure out why the rule for P -> Q does not apply to P -> ~Q.
9
votes
2answers
363 views

How does “higher-order logic” differ from “normal” (first order?) predicate logic?

How does “higher-order logic” differ from “normal” predicate logic? I assume the latter is consistently called ”first order logic”. So where are the differences between these? What kinds of statements ...
2
votes
4answers
642 views

What is the real world, mathematically?

Part and parcel of Plato's Platonic Realism is his theory of Forms or Ideas, which refer to his belief that the material world as it seems to us is not the real world, but only a shadow or a poor copy ...
5
votes
5answers
1k views

How to understand numbers that become really large?

If we begin with a notion of number N that we denote F(N) as a function of time, can a decidable procedure exist on definability of the growth of numbers? Inspired by Tipler's Omega point and Thomson'...
3
votes
2answers
2k views

Relation of Gödel's incompleteness theorems and Karl Popper falsification

Falsifiability is considered a positive (and often essential) quality of a hypothesis because it means that the hypothesis is testable by empirical experiment and thus conforms to the standards of ...
2
votes
2answers
203 views

What would Kant say, if I asked him about the ontological status of the integers?

I understand that Kant remarked that Space & Time are forms that the intuition take. Would he also say that of the integers? Are they judgements? That is they lie within his Category of Quantity.
4
votes
5answers
623 views

If I am infinitely old , can I have a father?

If I am infinitely old , can I have a father ? And can I have a brother that is infinitely older than me but younger than my dad ?
5
votes
3answers
764 views

How could the concept of 'evidence' be defined, and how significant is it?

What is evidence, and how much of it means that a proposition is true? Does a partial / total lack of evidence mean that a proposition should be ignored? Is the concept evidence more important to ...
2
votes
1answer
360 views

Is mathematics infinitely regressive?

Agrippas trilemma states that formal systems are either self-circular, infinitely regressive or axiomatic. Its commonly taken that mathematics is axiomatic. However just as mathematicians can build ...
6
votes
3answers
214 views

What's a name for the impossibility of identity?

It appears to me that no two things can ever be identical, yet the notion that they can has been deployed rather without pause about a billion times in theoretical literature in philosophy and ...
0
votes
1answer
576 views

What's the relationship between infinity and a dimension? [closed]

When I was reading Kant's Critique, I got the sense that he'd sort of found a formula for calling something a dimension. Space seems to arise out of an infinity of extension. Time seems to arise ...
7
votes
3answers
314 views

A (possible) puzzle regarding John Lane Bell's “Abstract Sets”

John Lane Bell, is his paper "Abstract and Variable Sets in Category Theory" (go to Bell's Homepage to download it), defines an abstract set as follows: "An abstract set is then an image of pure ...
1
vote
1answer
144 views

The represenation of nothing

If zero is the representation of nothing, then nothing must me something because it is being represented, correct? Now, if the above is incorrect, and zero is actually nothing, then why is it that ...
2
votes
2answers
241 views

Which other philosophers other than Aristotle have discussed the continuum?

Aristotle claimed that the continuum, that is say a line, is not solely composed of points. Modern mathematics would agree, they additionally impose a topology to achieve cohesion. Have their been ...
3
votes
1answer
216 views

Are numbers noumena?

According to OED, noumenon is An object knowable only by the mind or intellect, not by the senses But I'm a little confused at considering about numbers, they seem to be objects knowable only by ...
4
votes
3answers
268 views

Expressing identity in mathematics

I considered asking this on math.SE, but I realized this question wasn't really about mathematics. Suppose I have 4 pens sitting on my desk. So I have a set S = {pen, pen, pen, pen} = {pen}. That's ...
6
votes
2answers
281 views

What is the ontological stance of formalists on mathematical objects?

Are modern proponents of formalism associated with an ontoglogical opinion regarding numbers? If they view mathematics as the process of manipulating string according to agreed upon rules, there is ...
2
votes
2answers
405 views

To what extent did belief in monotheism play a role in the development of modern probability theory?

The most appropriate statement of monotheism is surely the Shema which, as is widely known, is highly related to the Abrahamic faiths. In Chapter 1 of the book "against the gods: the remarkable story ...
0
votes
1answer
120 views

Cognitive science/brain sciences and their impact on philosophy of mathematics

How does/did cognitive science influence philosophy of mathematics? I saw somewhere (Wikipedia, "Cognitive science") that it helped to create new perspective on philosophy of mathematics, but it did ...
1
vote
2answers
364 views

What are the main philosophical investigations into the amazing ability of extremely complicated mathematics to model the physical world [closed]

or of our mental representation of it? Edit: Michael Dorfman raises the point, "why would it be amazing that "extremely complicated mathematics" would model the physical world? ... In fact, it would ...