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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.

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1answer
272 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 ...
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3answers
228 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 ...
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1answer
146 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, ...
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1answer
449 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 ...
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1answer
195 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 ...
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411 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 ...
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2answers
108 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 ...
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2answers
405 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: ...
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1answer
357 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 ...
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4answers
879 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. ...
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3answers
427 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 ...
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1answer
592 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 ...
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1answer
582 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 ...
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2answers
219 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 ...
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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 ...
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1answer
714 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 ...
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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 ...
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1answer
494 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.
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2answers
370 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 ...
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4answers
655 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 ...
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5answers
2k 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'...
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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.
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630 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 ?
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767 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 ...
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1answer
367 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 ...
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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 ...
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1answer
593 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 ...
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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 ...
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1answer
160 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 ...
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2answers
243 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 ...
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1answer
223 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 ...
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271 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 ...
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283 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 ...
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2answers
413 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 ...
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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 ...
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2answers
369 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 ...
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223 views

Genus-Differentia and Mathematical Categories

I am a mathematician by training. Category theory has become a major subfield of mathematics --- major enough that some have tried to recast the logical foundations of mathematics in terms of ...
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3answers
1k views

What is the philosophical problem with Skolem's Paradox?

I guess there are two questions here. QUESTION 1: Skolem's Paradox shows that countability is relative in first-order logic, but where is the relativity? In this first question, I will do the ...
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3answers
905 views

How do mathematical objects relate to the real world?

I am just going to give an example of what I mean using Skolem's Paradox. I DO NOT want to get into Skolem's Paradox itself or its "resolution." Skolem's showed that countability is relative in ...
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326 views

Can mathematical sentences in different theories be identified?

My question motivated by a part of this page from Saul Kripke's book Naming and Necessity, which is also viewable on google books. In the middle of the page he say something, which seems unnatural to ...
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547 views

Has there been a Cartesian revolution in mathematics?

In his book "Méthodes modernes en géométrie", Jean Fresnel wrote: il ne faut pas se faire d'illusions, Descartes résout des problème de géométrie, non parce qu'il a de la méthode, mais parce qu'...
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2answers
106 views

Does it make sense to think that algorithms can be specified only for all that which is manmade? [closed]

I've been asking myself the following question over and over again: can one write an algorithm (a series of steps for solving a problem) for something that came about through a process that is at ...
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3answers
429 views

Can something infinite be absolute? [closed]

Let's say we have an equation that has no end to its result. (Sorry I don't have an example to hand, and the value of Pi is still under question so I won't use that). Can this value be considered ...