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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190
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29answers
29k 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/...
108
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21answers
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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 ...
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25answers
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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 ...
45
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13answers
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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 ...
44
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15answers
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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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12answers
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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 ...
34
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3answers
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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 ...
33
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12answers
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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 ...
33
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13answers
5k 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 ...
32
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17answers
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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 ...
32
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3answers
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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 ...
29
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4answers
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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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3answers
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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 ...
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7answers
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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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4answers
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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 ...
23
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14answers
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What would happen if suddenly, 1+1=2 is disproved?

Would the universe be thrown into chaos when the most fundamental equation is proved to be wrong?
22
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6answers
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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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14answers
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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 ...
21
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5answers
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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 ...
21
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4answers
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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 ...
20
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6answers
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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 ...
20
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7answers
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If there were only one single mathematician in the world, would s/he be able to produce a mathematical proof?

If there were only one single mathematician in the world, would s/he be able to produce a mathematical proof? This question was motivated by the Math stackexchange question: Should a mathematical ...
20
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7answers
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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, ...
20
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14answers
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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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7answers
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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 ...
18
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7answers
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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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8answers
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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 ...
17
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13answers
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Is mathematics truth?

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 ...
17
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11answers
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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 ...
17
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8answers
1k 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 ...
16
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13answers
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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? ...
15
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9answers
458 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, ...
15
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7answers
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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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4answers
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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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9answers
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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 ...
14
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5answers
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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 ...
14
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1answer
327 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, ...
13
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5answers
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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 ...
13
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3answers
426 views

What is the relation between proof in mathematics and observation in physics?

Recently in his 2015 Hirzebruch Lecture in Bonn, Arthur Jaffe re-amplified his famous perspective that finding proof in mathematics is analogous to making experimental observation in physics. In ...
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4answers
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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 ...
13
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1answer
794 views

What sources discuss Russell's response to Gödel's incompleteness theorems?

In his book My Philosophical Development Russell writes, In my introduction to the Tractatus, I suggested that, although in any given language there are things which that language cannot express, ...
12
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4answers
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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 ...
12
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3answers
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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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2answers
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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 ...
12
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3answers
736 views

How come intuitive thinking is related to constructing a proof?

I am researching Constructivism and Intuitionism. I can't understand why Intuitionism and Intuitionistic Logic are named as they are. Intuitionistic logic requires constructing a proof of every ...
12
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8answers
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Does claiming something exists imply that the number 1 exists?

The number 1 is used in language when we make claims of existence concerning distinct well-defined objects. It seems then that to say the number 1 does not exist would imply that nothing exists at all....
12
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2answers
705 views

What is the philosophical status of category theory?

In the philosophy of mathematics, some attempts have been made to give it ultimately secure foundations; a notable example is the Hilbert Program. Goedel's Theorems show that it is not quite possible, ...
12
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1answer
555 views

What are the differences between Tarski's 1933 and 1956 truth definitions?

The paper "The Seven Virtues of Simple Type Theory" mentions that it uses the same trick (due to Tarski) to define the semantics that is also used by first-order logic. I interpreted this a reference ...
11
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6answers
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Quantum Mechanics and Logic [closed]

I heard several times that the results of quantum mechanics (double-slit experiment for instance ) challenge our logic. One example of that is the famous physicist Lawrence Krauss. He keeps ...
11
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9answers
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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 ...