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.
1,261
questions
220
votes
23
answers
47k
views
Was mathematics invented or discovered?
What would it mean to say that mathematics was invented and how would this be different from saying mathematics was discovered?
Is this even a serious philosophical question or just a meaningless/...
- 2,784
118
votes
22
answers
23k
views
Why don't fair coin tosses "add up"? Or... is "gambler's fallacy" really valid?
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" or whatever has "no memory."
At each toss the odds ...
- 13.4k
68
votes
29
answers
13k
views
Why is there something instead of nothing?
A simple but fundamental question.
The "something" means the whole Universe (known and unknown), it could be represented as the reality version of the set of all sets, which is itself debated. It ...
- 1,056
50
votes
15
answers
16k
views
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 ...
- 609
47
votes
4
answers
16k
views
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 ...
- 2,014
45
votes
13
answers
18k
views
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 ...
- 769
40
votes
13
answers
8k
views
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 ...
- 657
38
votes
11
answers
6k
views
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 ...
34
votes
12
answers
10k
views
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 ...
- 1,841
34
votes
3
answers
5k
views
How is Gödel's incompleteness theorem interpreted in intuitionistic logic?
Classically, one sets up an axiomatic system with a formal deduction system & an interpretation in a model. Generally it is sound, that is: a formally deduced theorem is also true when interpreted ...
33
votes
3
answers
4k
views
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 ...
- 606
32
votes
17
answers
14k
views
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 ...
- 834
32
votes
4
answers
16k
views
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
...
- 9,492
30
votes
10
answers
10k
views
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 eventually produce ...
- 411
29
votes
16
answers
10k
views
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 ...
- 5,648
27
votes
7
answers
1k
views
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, ...
- 373
26
votes
7
answers
5k
views
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 ...
- 667
26
votes
7
answers
49k
views
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 ...
- 2,890
26
votes
4
answers
5k
views
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 ...
- 371
25
votes
14
answers
5k
views
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 ...
user1207
23
votes
16
answers
37k
views
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?
- 435
23
votes
17
answers
14k
views
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 ...
- 526
23
votes
5
answers
3k
views
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 ...
- 2,673
23
votes
7
answers
8k
views
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, ...
- 331
21
votes
16
answers
20k
views
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? ...
- 399
21
votes
14
answers
6k
views
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. ...
- 354
20
votes
7
answers
7k
views
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 ...
- 318
20
votes
11
answers
23k
views
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 ...
20
votes
7
answers
3k
views
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 ...
20
votes
10
answers
4k
views
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 ...
- 303
19
votes
6
answers
7k
views
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 ...
- 475
19
votes
7
answers
6k
views
Selection of logical connectives {¬,∨,∧,⇒,⇔} in set theory?
Nearly every treatment of set theory, whether Paul Halmos' Naive Set Theory, Herbert Enderton's Elements of Set Theory, Patrick Suppes' Axiomatic Set Theory, etc. introduce a common set of logical ...
- 799
18
votes
6
answers
9k
views
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 ...
- 481
18
votes
13
answers
9k
views
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 ...
- 283
18
votes
8
answers
2k
views
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 ...
- 1,067
17
votes
9
answers
6k
views
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&...
- 5,648
17
votes
10
answers
13k
views
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 ...
- 317
17
votes
2
answers
2k
views
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 ...
- 2,673
17
votes
4
answers
3k
views
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 ...
- 2,673
17
votes
6
answers
3k
views
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 ...
- 295
17
votes
1
answer
545
views
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, ...
- 7,817
16
votes
16
answers
7k
views
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 ...
- 1,073
16
votes
3
answers
2k
views
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], ...
- 440
15
votes
9
answers
572
views
Is 'equality' ultimately grounded in empirical observation?
Let's say I invent a concept X in my own imaginings. The only property it has is X-ness; it is defined as 'that which is represented by X'. I have just defined that to be the case. It seems to me, ...
- 2,009
15
votes
9
answers
9k
views
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 ...
- 1,514
14
votes
7
answers
9k
views
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 ...
- 150
14
votes
5
answers
3k
views
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 ...
- 141
14
votes
2
answers
3k
views
What are the truth-values of intuitionistic logic?
Classical propositional logic is bivalent, that is its set of truth-values has cardinality 2 (True & False). Intuitionistic logic drops the law of the excluded middle; does it have the same set of ...
14
votes
5
answers
3k
views
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 ...
13
votes
3
answers
9k
views
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 ...
- 233