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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Is mathematics invented or discovered?
discovery = finding something that existed before, for the first time (e.g., a frog, black holes)
invention = intellectual creation of something that did not exist before (e.g. a system, concept, ...
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Why is the gambler's fallacy a fallacy?
I have always been perplexed by a seeming paradox in probability that I'm sure has some simple, well-known explanation
We say that a "fair coin" has "no memory."
At each toss, the ...
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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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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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Do numbers exist independently from observers?
Do numbers have an objective existence? If life had not evolved on planet earth would there be numbers or are numbers an invention of human minds?
Are there any relevant works that discuss this? (I ...
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What should philosophers know about math and natural sciences?
My question is whether a lack of knowledge about formal mathematics or theoretical science in general would have an impact on a philosopher's ability to think and make judgments.
Why should a ...
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Why is the complex number an integral part of physical reality?
In modern physics, the quantum wave distribution function necessarily uses complex numbers to represent itself. If physics defines the physical reality, then what we are saying by the statement above ...
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How does mathematics work?
If I am given a parking lot with ten thousand cars and I want to determine whether one of the cars is orange, the only way I can do this is go through the parking lot examining each car until I find ...
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Isn't the notion that everything will occur in an infinite timeline an example of the gambler's fallacy?
I've seen a few different formulations of this, but the most famous is "monkeys on a typewriter" - that if you put a team of monkeys on a typewriter, given infinite time, they will ...
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Is mathematics politically and culturally neutral?
Lately, there have been many people who say that mathematics itself is racist, that it is simply a creation of dead white Greek men. As a mathematician, I strongly disagree, and believe that ...
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What are the philosophical implications of Gödel's First Incompleteness Theorem?
Gödel's First Incompleteness Theorem states
Any effectively generated theory
capable of expressing elementary
arithmetic cannot be both consistent
and complete. In particular, for any
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What are some works that apply an axiomatic method to something other than mathematics?
The axiomatic method is today mostly associated with mathematics. However, historically there have been some works, as for example Spinoza's Ethics, that have applied axiomatic method to philosophy, ...
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What is the difference between a statement and a proposition?
I'm doing a MOOC on mathematical philosophy and the lecturer drew a distinction between a proposition and a statement. This is very puzzling to me. My background is in math and I regard those two ...
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Why is Aristotle's objection not considered a resolution to Zeno's paradox?
It seems to me, perhaps naïvely, that Aristotle resolved Zenos' famous paradoxes well, when he said that,
Time is not composed of indivisible nows any more than any other magnitude is composed of ...
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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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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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What would happen if suddenly, 1+1=2 is disproved?
Would the universe be thrown into chaos were the most fundamental equation proved to be wrong?
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Is infinity a number?
So I've been on a number of math fora, part of learning some calculus (not much of set theory, no). To my surprise I found what I would describe as strong resistance from some folks against (using) ...
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Are numbers real?
I am confused as to what numbers are. Numbers are defined to be what they are, so numbers aren't real? But numbers are found in nature, right? So if we invented them, how can they be found in nature? ...
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What are the foundations of philosophy?
I'm a student majoring in mathematics. I've taken a course in mathematical logic and a course in set theory. My problem is basically that I'm always finding philosophical concepts, for example syntax, ...
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Is mathematics truth? As in the sense of that which is manifest or possible in reality?
In mathematics there are imaginary numbers which cannot be represented directly in reality (the physical world). For example, you can't have i apples where
i = √-1 (square root of -1)
Can we ...
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Does Math use the scientific method?
I've reading many entries about whether Math uses the scientific method and the dominant opinions seems to be "no", e.g. from "Is Mathematics a science?" and other websites.
James ...
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What do all branches of Mathematics have in common to be considered "Mathematics", or parts of the same field?
At some point in my life I think I've read what all branches of Mathematics had in common were numbers. But then I remembered a branch of the many Mathematics I had when I was an university student, ...
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What was Cantor's philosophical reason for accepting the infinite but rejecting the infinitesimal?
I have begun inquiring recently into mathematical aspects of Georg Cantor's theory of transfinite numbers and sets, which he developed between the years of 1874 and 1897. Throughout his theory, Cantor ...
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Is Mathematics considered a science?
Science, generally is analyzing information gathered from observing phenomena, and coming up with theories to try and explain the phenomena. Then, attempting to predict a new phenomenon before it ...
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What is a straight line?
I am not a philosopher; I am an engineer with a reasonable grasp of mathematics.
This question has been bothering me for a long time, and I have asked a variation of it to a mathematical community. ...
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Why would infinite monkeys not produce the works of Shakespeare?
Apologies if this is a very basic/obvious question. I have no training in philosophy, but have been making my way through Peter Adamson's History of Philosophy podcast.
Recently I listened to his ...
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If there were only one single mathematician in the world, would they be able to produce a mathematical proof?
If there were only one single mathematician in the world, would they be able to produce a mathematical proof?
This question was motivated by the Math stackexchange question:
Should a mathematical ...
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What is the philosophical ground for distinguishing logic and mathematics?
I was wondering why the field of mathematics and that of logic are perceived as two distinct fields. Although could be pleased with the intuition that logic is rather meta-mathematics, still would ...
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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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Interpret Bayesian probability as frequentist probability?
It is usually said that the Bayesian probability is a subjective concept, quantifying one's degree of belief in something, while the frequentist probability is the the fraction of certain outcomes ...
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Falsification in Math vs Science
In the beginning it was thought that the statement 1+1=0 is false, and necessarily so.
However, with the birth of modular arithmetic, it was found that indeed, 1+1 does indeed equal to 0 (in the mod 2 ...
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Is there a notion of "because" in mathematics?
Sometimes, in math classes, we are asked to give justification for our mathematical assertions. We say that mathematical statement X is true because Y is true. However, I don't know if "because&...
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Which comes first - truth or provability?
When I'm thinking about mathematics, I usually imagine that every sentence in the language of arithmetic is either true or false, in reality. Thus, I imagine that truth comes first. Afterwards come ...
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Why can't numbers be 'used up'?
I was speaking with a young student who has been learning about addition and subtraction (essentially functions, but he doesn't know that yet) with the idea of a 'number machine' and he could not ...
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If math is so deductive, why is it so hard to discover new math?
Some considerations:
The conclusions of much latter/new math may be said to be already existent within the premises of current math
The importance of deduction changes depending on if math is said ...
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What is a natural number?
It’s been on my mind lately. I do maths and work with them daily, but I’m not entirely sure of what they really are.
I understand they are symbols at a surface level, but there is obviously more to it....
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What is the idea behind "p or not p" being a tautology?
Most (all?) logic books consider "p or not p" to be a tautology, hence always true, and this is usually stated without any further discussion. (I never gave it a second thought.)
In common ...
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Why were Kant's categories used in the mathematical category theory?
I am curious exactly how mathematical categories were inspired by Kant's categories. The SEP article on category theory says:
In order to give a general definition of the [natural transformation], ...
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Are mathematical suppositions of physical theories determined uniquely according to Aristotle and Plato?
Does mathematics apply to physics in one way or multiple ways? What do Aristotle and Plato think?
It would seem that Aristotle thinks mathematics can be applied to
physics in one way only because, ...
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Is geometry mathematical or empirical?
Is Euclidean geometry a mathematical theory, or is it a theory of empirical science?
If taking it to be a mathematical theory would it be due to having alternative geometries? If so, is it in some ...
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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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Is Logic Empirical?
We use the logical system that we know from observations (empirical data) holds true in the world we live in (please correct me if I am wrong). Hence the axioms of logic we choose are themselves ...
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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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Is the surprising applicability of mathematics to the physical world a brute fact, or something crying out for a theistic explanation?
William Lane Craig proposed the following argument for God's existence:
For those who are unfamiliar with the argument for God from the applicability of mathematics to the physical world, here is a ...
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Why do universities not teach constructive mathematics to CS undergraduates?
I had a conversation with a user on the Internet. And it did indeed wake my interest regarding something that I had also been asking myself long ago. Why do so many universities still teach beginners ...
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Is it possible for everything that exists to have a definition?
Is it possible for everything that exists to have a definition? I actually started out asking this in the linguistics - semantics stack and was directed here. By definition I mean at least in the ...
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Thomson's lamp: a useless paradox?
Thomson's lamp was mentioned at How to understand numbers that become really large? (as well as a number of posts elsewhere on SE). I have mentioned elsewhere that in addition to Cantorian infinite ...
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What is the difference between an Ordinal number and a Cardinal number?
I'm trying to understand the real difference between an Ordinal and a Cardinal, especially in relation with transfinite cardinals. The stuff on Wiki is a bit too complicated. Can anyone make it simple ...
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Is a proof still valid if only the author understands it?
Some time ago I was reading about the recent Shinichi Mochizuki's proof for the famous ABC conjecture. It's enormous and so incredibly difficult that at that time virtually nobody was able to ...
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