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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314 views

How was Edmund Husserl viewed among the mathematical community in the late 19th/early 20th centuries?

How was he viewed. I am interested in what the mathematical community thought of his philosophy of mathematics. What Hilbert thought of him etc. etc.?
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What does the term “mathematical logic” mean?

What is "mathematical logic"? Is it the logic of mathematical reasoning, or is it the claim that mathematics and logic are identical? Also, is "quantificational logic" a particular type of "...
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Does phenomenology determine ontology?

There were many historical instances where phenomena could be explained by seemingly incompatible theories, Copernican and Ptolemaic systems, corpuscular and wave theories of light, interpretations of ...
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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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173 views

Poincare says we are born geometric or arithmetic thinkers. Which was Grothendieck and why?

Poincare proclaims that the mathematical continuum originates from the sensible intuition and that intuition by pure number or logic alone could not have given us this notion. Source for the claim: ...
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Do mathematicians use logic when adding things up?

I asked a similar question recently, and it was closed. So, bear with me. When two things are materially equivalent we don't add anything to work out how much we have of both together, right? If ...
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143 views

How is ω-consistency different from ordinary consistency?

I've read Gödel's explanation and others but my understanding is unclear. Answers to the followup questions below would help: does ω-consistency have any relevance to methods or ideas not connected ...
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What is the difference (if any) between the concepts of natural numbers and finite cardinals?

The definition of natural numbers from Wikipedia: In mathematics, the natural numbers are those used for counting (as in "there are six coins on the table") and ordering (as in "this is the third ...
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166 views

Is symbolism necessary in mathematics?

I have a penchant of writing mathematical pieces (proofs, theorems, axioms, etc.) in formal syntax, using as less English as possible. I don't know whether this is essential, I like doing this so I do ...
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What is the intersection of Physics & Philosophy?

(I've already asked this question on Meta, but as one answer (by Joseph Weissman) pointed out this is already a philosophical question; so I thought it worth asking here). I've asked a number of ...
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215 views

What philosophical axes did 19th century mathematicians have to grind?

Tim Button's presentation of set theory motivates the subject by providing a history of 19th century mathematics where the notion of limit allowed definitions of the derivative and continuity. These ...
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244 views

What does Wittgenstein mean when he says “there are no numbers in logic”?

From the Tractatus: 5.453 All numbers in logic stand in need of justification. Or rather, it must become evident that there are no numbers in logic. There are no pre-eminent numbers. What does ...
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Are infinitesimals in the Newton and Leibniz calculus potential or actual?

Was reading a bit about history of calculus and its philosophy and stumbled into source of confusion: were infinitesimals in seventeenth century calculus assumed to be actual or potential? Was there ...
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Logic of inductive inference will free statisticans - why?

I'm doing my best try to understand this excerpt of Efron's article (1998) on Fisher: Fisher believed that there must exist a logic of inductive inference that would yield a correct answer to any ...
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What is mathematics? [duplicate]

Is mathematical practice: an act of discovery of eternal objects and ideas independent of human existence; an intuition-free game in which symbols are manipulated according to a fixed sets of rules; ...
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Is the number of universes finite, countably infinite or uncountably infinite (and what size of uncountable if so)?

Assuming that the alternative universe theory is correct, how many alternate universes are there? From my understanding an alternate universe "pops up" whenever a particle goes from being in an ...
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473 views

Are there theories of arithmetic that are inconsistent with the natural numbers?

The programme of ultrfinitism dispenses with the notion of very large finite numbers simply becaause they argue that such large finite numbers have no way of being conceptualised in our universe in a ...
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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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428 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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Is time more “real” than math and, if so, why?

How is time different from math? Is time a part of math? For me time is like math rather than a "real" thing. Time is just a tool rather than a "fundamental thing". I feel confident saying that an ...
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134 views

What do we mean by philosophy of physics or math what does it tell us about them?

what do we mean by philosophy of physics or math what does it tell us about them ? Does it tell about how these disciplines of knowledge develop? For example to develop math we use axiomatic approach ...
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239 views

Do mathematical entities transcend duality and cause/effect?

In the Wikipedia entry on Philosophy of mathematics, the following is mentioned about Platonism: [M]athematical entities are abstract, have no spatiotemporal or causal properties, and are eternal ...
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387 views

Justification of implication-introduction and modus ponens

Given only the definition of material implication through the truth table A | B | A → B ------------------ f | f | t f | t | t t | f | f t | t | t (where, as usual, "t" ...
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201 views

How would Hume classify computer generated mathematical proofs?

Hume's fork divides knowledge of the world into: Analytic a priori: relations of ideas. Synthetic a posteriori: matters of fact, empirical statements about the world. How would Hume classify ...
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616 views

Is there a One True Set Theory?

From the description of Category Theory in nlab: Category theory is a structural approach to mathematics that can (through such methods as Lawvere's ETCS) provide foundations of mathematics and (...
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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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Can a physicalist be also realist about mathematical objects?

Is it possible to believe that mathematical objects enjoy some kind of mind-independent existence while holding physicalism? And if they are mind-dependent, should one embrace constructivism ...
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292 views

Ontological status of Axiom of Choice

Mathematical facts are necessary truths, either in a Platonic sense or by way of axioms. In the latter sense I mean that the Peano Axioms prove that 2+3=5, for example. In other words, "PA ⊨ 2+3=5"...
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Introduction to science philosophy

I'm interested in studying philosophy as a framework for thinking about engineering. I have a background in science. Who are some interesting thinkers for science, engineering and math? More ...
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574 views

What do logicists mean when they try to “reduce mathematics to logic”?

I've read a lot about Russell and other Logicism advocates and their trial to reduce math to logic. But what does that mean? We know that all known mathematics can be reduced to Set theory, is that ...
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How should one properly characterize mathematical conclusions?

I am a mathematics graduate student, not a philosophy student, so please bear with me. However, I am interested in investigating what exactly it is that I spend the majority of my week doing! As ...
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410 views

Does one have to become a Platonist to refuse to be a Platonist?

I believe the answer is no, but Scott Aaronson on his blog just gave in interesting argument to the contrary. This is in connection with the now famous paper Undecidability of the Spectral Gap, and ...
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384 views

Why must we choose an intuitionistic explanation over a paraconsistent one, given they are dual?

Given the anti-intuitive results of Quantum Mechanics, it is not surprising that Physicists would look for a deeper reason in the structure of the theory to explain what was then (and still is) ...
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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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574 views

Gödel: Why is a proposition undecidable?

Gödel has proved the existence of undecidable propositions for any system of recursive axioms capable of formalizing arithmetic. But do we know the logical causes of this state of undecidability? In ...
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388 views

In what sense is set theory the “meta theory” of analysis?

Here, Terence Tao said: I deliberately chose not to be excessively formal with regards to the set-theoretic foundations of mathematics here, which I am regarding as part of the metatheory of ...
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135 views

Are all the consequences of a science in the science's principles?

"Chaotic" differential equations are very simple principles compared to the more complex consequences of them. For example, the equations modeling the motion of a double-pendulum, ,are relatively ...
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511 views

difference between conception and theory

What is the difference between a conception of something and a theory of something? Is the conception more extensive in content or less? Example: iterative conception of sets and ZFC
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The notion of knowledge for Kant and mathematical objects

As far as I understand the notion of knowledge in Kantian philosophy, we cannot speak of knowing something unless there is a relation between its concept and some object of intuition in experience. ...
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What do we mean by the term “Number of things”?

I am reading the book "The Number-System of Algebra (2nd edition)." I have some problems with the the first article: "Number". The author has confined the concept of number of things to the groups ...
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Is there an alternative to Cantor's cardinalities that makes proper subsets smaller than their sets?

Cantor defined an infinite set as a set whose subset can be placed in a one-to-one correspondence with its subset. That is, take the set of all natural numbers: {0, 1, 2, 3, 4,...}. From that set, you ...
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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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Do transfinite sets have practical applications?

This may not qualify as a philosophy question exactly, but I would argue that potential applications of pure mathematics are in the bounds of philosophical interest. Many innovations in pure ...
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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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How close is philosophical thinking to mathematical thinking?

I made this question because I've seen so many mathematicians who were also philosophers and vice versa; also from the way that mathematicians can build arguments in non-mathematical contexts, which ...
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851 views

Opposite Universe

Let us assume we live in a shared Universe which can be fully described mathematically, and that all mathematical variables have an opposite. 2 is opposite to -2, and true is opposite to false. When I ...
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Is logic subjective?

If logic is constructed from axioms, and axioms are depended on observation which in term could be subjective, does this means that logic could be limited to our observation, and not really absolute ...
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853 views

Why are set theory and numbers important to philosophy?

I'm reading David Papineau's Philosophical Devices, and there's a section on numbers and set theory. But there's not deeper hint on why it's important to philosophy. I guess that in mathematics we ...
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The belief that everything is expressible in mathematical terms?

For want of a better word, mathematicism will be defined as the belief that everything is expressible in mathematical terms. I'm not sure if this is a position that anyone affirms, as my thoughts on ...
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GUT and TOE as Fallacies of Misplaced Concreteness?

A.N. Whitehead warns in the introduction to Process and Reality, that the “chief error” of Western philosophy is “overstatement.” He states: “the aim at generalization is sound, but the estimate of ...