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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Cantor and infinities
I know we have accepted Cantor's ideas a long time ago and many mathematicians use sets and infinities without ever realizing that thinking about sets and infinities intuitively fails, because there ...
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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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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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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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Did Russell understand Gödel's incompleteness theorems?
Russell was active in philosophy (although no longer in math) for many years after the Gödel's 1931 publication. Gödel's paper were not obscure, and Russell would have been aware of their effect on ...
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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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Is there a known limit to relationship between physics and mathematics?
I am much interested in discussions such as Wigner's "The Unreasonable Effectiveness of Mathematics in the Natural Sciences". It's quite amazing that mathematics so well applies to our universe, and ...
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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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Has there been any philosophical investigation into the role of aesthetics in mathematics?
There are many mathematicians who talk about the particular beauty of a subject. They may say a particular result is pretty. It may be beautiful.
It seems to me play a fundamental role in the ...
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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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Is Mathematics always correct?
It seems Mathematical theories/Laws/Formulas are the least questioned in all of the sciences. Is mathematics that good at being closest to the laws of universe, or is it just a logical tool of our own ...
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Do the laws of logic exist independently of human or animal consciousness?
Are the laws of mathematics and logic, such as if a=b, and b=c, then a=c just constructs of the human mind, or does the universe hold an innate logical structure to it, which the physical part of the ...
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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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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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Cosmological truth and the theory of cosmic epochs
According to Whitehead’s theory of cosmic epochs and extensive continuum, there are material conditions that drive the more general metaphysical descriptions and principles. We only can account for ...
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If the universe is infinite, shouldn't I already have been contacted by a time and space travelling doppelgänger?
If the universe is infinite, by virtue of chance it means that every possible configuration of matter must exist somewhere (according to this documentary).
Therefore, if the universe is infinite and ...
user2215
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Does a Background in Mathematics Make One a Better Philosopher?
I was a Philosophy major as an undergrad and became obsessed with the beauty of rigorous argumentation. There I didn't take a single class listed under the Mathematics department and was almost ...
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Is First Order Logic (FOL) the only fundamental logic?
I'm far from being an expert in the field of mathematical logic, but I've been reading about the academic work invested in the foundations of mathematics, both in a historical and objetive sense; and ...
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What is the dimension of a curved plane? [closed]
A plane is defined by being 2-dimensional. But if it is curved (like a hyperbolic plane), it requires an extra dimension, that is, it is curved in a 3-dimensional space.
So it seems, 2-D coordinates ...
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What are the philosophical implications of the Halting Problem?
In a great answer, a community member gave the following proof sketch that the halting problem is undecidable:
Proof that the halting problem is undecidable. If there were a computable procedure to ...
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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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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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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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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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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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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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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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"Because if you doubt that you're doubting, you're still doubting" - What is the analogous mathematical/logical expression to this sentence?
In an answer here, the following was stated:
The essence of his [Descartes] argument is that you can doubt almost everything about the world, but you can't doubt that you're doubting. Because if ...
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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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Can one still derive paradoxes from the amended version of Naive Set theory given by Cantor in a letter to Dedekind?
Consider the following definition of set given by Cantor in a letter to Dedekind:
If on the other hand the totality of the elements of a multiplicity can be thought of without contradiction as 'being ...
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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 ...
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Do Gödel's incompleteness theorems support the idea that the examination of a 'system' should only be undertaken to arrive at the inconsistency?
Roughly, Gödel demonstrated that in a logical system, that contains a model or arithmetic, there are statements which may be true, but are unprovable within the system.
If a statement is not ...
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Are there any known deficits of "relevant logic"?
The principle of explosion is the law of classical logic and similar systems of logic, according to which any statement can be proven from a contradiction. Some early formal systems like Frege's ...
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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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To what extent can the invention of zero in India as a number be tied to Buddhist philosophy, if at all?
The Wikipedia entry on zero suggests that the ancient Greeks were unsure about the ontological status of zero. They asked themselves, 'How can nothing be something?' whereas in Buddhism, Sunyata or ...
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What are the philosophical implications of category theory?
I have heard about topoi being the ideal entities to use for foundations of mathematics (since we are able to reasonably interpret our theories in them), so I imagine there might possibly be some ...
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What is a mathematical or logical name for the process of proving a statement by exhausting the problem domain?
I am trying to understand logic and I came across a set of actions that I describe below that I can't get my head around.
Suppose you have a bag of multiple colored balls.
Situation 1.
Argument: ...
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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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Is Cantor's theorem based on a fallacy?
The Brazilian philosopher Olavo de Carvalho has written a philosophical “refutation” of Cantor’s theorem in his book “O Jardim das Aflições” (“The Garden of Afflictions”). Since the book has only been ...
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Why would Wittgenstein say we can't have a perfect language?
I have been reading Wittgenstein's Philosophical Investigations and my question is how does he come to realize that we can't have a perfect language.
For instance I would say math is a perfect ...
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Is mathematics founded on beliefs and assumptions?
Note: I originally posted the question in meta.math.stackexchange.com but I reckon this would suit a more philosophical audience so I am posting it here.
Background: I am a 28 year old ...
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How do we know how to follow a rule?
This question seems to either be at the forefront or the background of countless philosophical enquiries. Much has been written on Wittgenstein's rule paradox (e.g. Kirke's Wittgenstein: On Rules and ...
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How should we characterize the relationship between mathematics and philosophy of mathematics?
How should we characterize the relationship between mathematics and philosophy of mathematics?
Specifically, in what ways might the study of philosophy of mathematics make a mathematician better at ...
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What is mathematical existence?
When I make a claim in a proof that a mathematical entity exists, is this no more than saying that the theory I'm working within is consistent, and that all the steps upto that point in the proof are ...
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How can we have a 'natural existence' for complex numbers? [closed]
For those who don't know what a complex number is, in simple terms, a complex number is the square root of a negative number! For example, the square root of -1 is called a complex number. Those ...
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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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Is it possible to generate logically valid sentences made up of "atomic contradictions"?
Is it possible to generate sentences that are made up of "atomic contradictions", but which remain logically valid as a whole? By "atomic contradictions", I mean atomic propositions that are not ...
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What is a mathematical representation of the physical world?
Possible Duplicate:
Was mathematics invented or discovered?
To be more precise, does mathematics describe the physical world or does it describe a mental representation of the physical world? If ...
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Is illogical = not logical?
I think law of excluded middle makes sense to mean that a statement should be either logical or illogical but in this case I don't assume "not logical" = "illogical" since the author didn't say "...
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What are the necessary conditions for an action to be regarded as a free choice?
A common philosophical question revolves around the existence of free will, but what I've found is that these debates seem to gloss over the concept of "free will" itself, either taking it as a given ...
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