Questions involving philosophy of mathematics

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-3
votes
0answers
51 views

Language of science [on hold]

I need help discovering what this means - I wanna use mathematics terminology in everyday life for a certain purpose - But I perceive mathematics and reality as two separete entities - help!!!
-2
votes
2answers
144 views

Is Infinity Needed in Maths? Does Infinity Actually Exist? [on hold]

I'm asking this question as I have been having an on going online debate with a friend of mine. I claimed that Infinity does in fact exist in Maths and in Reality, as there's a whole plethora of ...
3
votes
5answers
142 views

Is $'' \sum_{n = 1}^{\infty} (-1)^n \; \text{is a real number}''$ an invalid statement or a false proposition?

So we're beginning an introductory logic course and my professor is giving examples for valid statements/ propositions - meaningful statements that are either true or false but not both. So he puts ...
13
votes
2answers
502 views

Proofs from the “Ugly Book”

There is a famous saying in mathematics from Paul Erdős: "You don't have to believe in God, but you should believe in The Book." "The Book" is an imaginary book in which God had written down the best ...
5
votes
2answers
47 views

How is Cartesian coordinate system related to his philosophy

In 1637, Rene Descartes published his famous monograph about philosophy "Discourse on the Method of reasoning well and Seeking Truth in the Sciences", and analytic method of geometry has been come up ...
121
votes
24answers
10k views

Can a coin with an unknown bias be treated as fair?

This morning, I wanted to flip a coin to make a decision but no coins were in reach. There was however an SD card on my desk: Given that I don't know the bias of this SD card, would flipping it be ...
-2
votes
0answers
93 views

Absoluteness of $\forall x (x=x)$

Is there any kind of (set-theoretic) absoluteness result for the formula $\forall x (x=x)$? And what about for $\exists x (x=x)$? I know $x=x$ is absolute given that it's $\Delta_0$. Also, if I'm ...
5
votes
2answers
638 views

Can one rigorously define “meaningful” versus “arbitrary” in math?

Often we regard certain mathematical expressions, or elements thereof, as arbitrary, in the sense that they have no apparent reason or cause, whereas more beautiful or natural seeming expressions feel ...
21
votes
3answers
1k views

What did Gauss think about infinity?

I have someone who is begging for a conversation with me about infinity. He thinks that Cantor got it wrong, and suggested to me that Gauss did not really believe in infinity, and would not have ...
1
vote
2answers
119 views

Three-valued logic as foundation

Isn't it more natural to use Three-valued logic(false-true-unknown) as the foundation of mathematics? It is a better model for natural languages. And it also can model sentences like the lair paradox ...
1
vote
1answer
23 views

Probability that theoretical results match experimental results

I am not sure if this can be determined, but I was wondering if there was any way to go deeper into probability to find the odds that your experimental results match your theoretical results. For ...
270
votes
35answers
27k views

Do complex numbers really exist?

Complex numbers involve the square root of negative one, and most non-mathematicians find it hard to accept that such a number is meaningful. In contrast, they feel that real numbers have an obvious ...
7
votes
3answers
134 views

The nature of infinities

I have been thinking about the nature of infinity lately. I have no experience with higher mathematics or theorems regarding infinity, so please forgive me if my ideas on this topic are extremely ...
3
votes
3answers
159 views

Provocations on the existence of mathematical objects

The few Mathematics I have been studying so far is pure Mathematics. I happen to have some discussions with philosophers of Mathematics, but as they know I totally ignore their subject, we do not ...
3
votes
2answers
132 views

Gödel's incompleteness theorems

In the last paragraph of Stephan Hawking's speech "Godel and the End of the Universe", he mentioned "... I'm now glad that our search for understanding will never come to an end, and that we will ...
2
votes
7answers
438 views

Are the real numbers really uncountable?

Consider the following statement Every real number must have a definition in order to be discussed. What this statement doesn't specify is how that loose-specific that definition is. Some examples ...
4
votes
4answers
380 views

The standard role of intuitive numbers in the foundations of mathematics

In my career I've been formed mostly in the formal side of mathematics, that is, standard set theory and every classical branch of mathematics that uses set theory. However, I am not quite sure about ...
1
vote
0answers
33 views

Why is time important in the Ross-Littlewood paradox?

I have read many defferent versions of the Ross-Littlewood Paradox. This post: Fun quiz: where did the infinitely many candies come from? This post: Paradox: increasing sequence that goes to $0$? ...
5
votes
0answers
45 views

Codifying ways to think and write about imprecise ideas?

This question will not be about affine spaces; rather those are mentioned here as one of many examples. A vector space has an underlying set and a field of scalars and an operation of linear ...
14
votes
8answers
1k views

Reference request: is mathematics discovered or created?

I have to write a short monograph as an assignment for a course on the philosophy of science. Being a math student, of course I want to opt for something math-related. After some initial ideas which ...
1
vote
2answers
144 views

What is the necessary condition for the process of “proceeding to the limit” in Whitehead and Russell's PM?

I read this from Introduction of the 1st edition of Principia Mathematica by Whitehead and Russell: Since the orders of functions are only defined step by step, there can be no process of ...
2
votes
1answer
45 views

Geometries (Euclidean and Projective)

We can think of Euclidean Geometry and Cartesian (Coordinate) Geometry as equivalent, in the sense that some proposition is true in Euclidean Geometry iff it's true in Coordinate Geometry. It makes ...
13
votes
4answers
218 views

Which mathematical ideas most influenced the way you think?

This is not a question about how you use a formula or mathematical method to solve quantitative problems - that is applied mathematics. Rather, I'd like to hear how deeper ideas gained through the ...
1
vote
3answers
88 views

Why Maximize Expected Value?

In many instances I've come across (in Game Theory, etc), when trying to choose an optimal strategy it has the criterion that it wants to maximize expected value much of the time. To simplify this ...
6
votes
6answers
260 views

Interviews of famous modern mathematicians

I was wondering, are there any good collections of interviews of famous modern mathematicians? It can be text interviews, or audio or video recordings. I am not sure what exactly I mean by "modern". ...
47
votes
10answers
12k views

How is a system of axioms different from a system of beliefs?

Other ways to put it: Is there any faith required in the adoption of a system of axioms? How is a given system of axioms accepted or rejected if not based on blind faith?
0
votes
2answers
116 views

Is the “Most Important Property a Set S has” Necessary and Sufficient to Define a Paradox-Free Notion of Set?

About a year and a half ago, while I was looking on the Web for papers regarding the Russell paradox, I chanced to find an interesting concept. This concept was contained in what (for want of a ...
1
vote
1answer
60 views

Formulation VS Interpretation

I'm reading a book on Mathematical Physics and at some point the author says that we must distinguish between a "formulation" and an "interpretation" of a theory, although it's not easy to point what ...
5
votes
1answer
218 views

What do ultrafinitists think about Graham's number?

I know ultrafinitists want to require not only that mathematical objects be constructible, but be constructible given finite resources (such as time). So I wonder about something like the famous ...
1
vote
3answers
73 views

Zermelo–Fraenkel set theory the natural numbers defines $1$ as $1 = \{\{\}\}$ but this does not seem right

If 1 can be defined as the set that contains only the empty set then what of sets which contain one thing such as the set of people who are me. number 1 does not just mean $1$ nothing, it means $1$ ...
14
votes
1answer
311 views

What lessons have mathematicians drawn from the existence of non-standard models?

So, as someone whose knowledge of mathematics has always come from studying it with an eye towards philosophical/foundational issues and studying it with other philosophers (who are not primarily ...
2
votes
1answer
102 views

Why didn't Frege succeed in his attempts to reduce mathematics to logic?

My background: Sophomore-level understanding of mathematics and philosophical logic. All the explanations I have found online so far are either far too technical or too simplistic. Thanks in advance ...
0
votes
0answers
28 views

Theoretical question of physical analogies to different O(f(x)) based characteristics of algoritms

I want to better understand the following concepts: "n!", "e^n". I.e. what is the physical analogy of the functions at the bottom of the message. F.ex. for the "n^a" and "log a x" where a equals to ...
3
votes
2answers
153 views

What is the meaning/purpose of finding the “foundations of mathematics”?

I've read in a lot of places how there was a "foundational crisis" in defining the "foundations of mathematics" in the 20th century. Now, I understand that mathematics was very different then, I ...
0
votes
1answer
56 views

particular property and completeness?

I was puzzeling with the almost standard definition of completeness: In mathematical logic and metalogic, a formal system is called complete with respect to a particular property if every formula ...
2
votes
3answers
289 views

How Do You Know If Mathematical Definition Matches Up With Reality?

This is probably one of the biggest question I have when learning some mathematics. I always wonder if I have a concept in my head lets say continuity. Lets I want this concept to be able to ...
1
vote
1answer
47 views

Formalized philosophy

I once recall a conversation with a friend who told me that his friend was taking a philosophy course where the ideas and concepts were formalized and done very rigorously. This really intrigues me. I ...
11
votes
0answers
486 views

Is there any formal definition or reasonably good heuristic for mathematical 'interestingness?' [closed]

One of the projects I'd like to work on over the next several years in my spare time is a first order theorem prover similar to Prover9 to attack some of the TPTP problems, and it occurs to me that ...
6
votes
6answers
2k views

Michael Spivak in “Calculus” asserts that $\sqrt2$ cannot be proven to exist, and that such a proof is impossible. What does he mean by “exist”?

Michael Spivak in "Calculus" asserts that $\sqrt2$ cannot be proven to exist, and that such a proof is impossible. What does he mean by "exist"? How are you to prove that any number "exists"? Why ...
43
votes
16answers
4k views

Is a proof still valid if only the writer understands it?

Say that there is some conjecture that someone has just proved. Let's assume that this proof is correct--that it is based on deductive reasoning and reaches the desired conclusion. However, if ...
26
votes
8answers
2k views

Complex analysis is more “real” than real analysis

In physics, in the past, complex numbers were used only to remember or simplify formulas and computations. But after the birth of quantum physics, they found that a thing as real as "matter" itself ...
0
votes
1answer
43 views

Zero vs Infinity relation type

I'm not sure it should be asked here or in philosophy. Bertrand Russell in his book "Introduction to Mathematical Philosophy" in chapter 7 when discussing rational numbers on page 66 says: "It will ...
2
votes
0answers
45 views

Why are divergent Fourier series all so 'HARD'?

I'm not sure if this question is appropriate or even making sense, but I still feel curious: why are every example of divergent Fourier series SO COMPLICATED? It usually takes pages to construct and ...
36
votes
14answers
10k views

How big is infinity?

This might be more philosophy than math, but it’s been bothering me for a while. Question: If there’s an infinite amount of real numbers between $ 0 $ and $ 1 $, shouldn’t there be twice the ...
10
votes
3answers
634 views

What is more important in Mathematics, Theorems or its Proofs?

Felix Klein once said, Mathematics has been advanced most by those who are distinguished more for intuition than for rigorous methods of proof. Till now I thought the opposite. I thought that ...
3
votes
1answer
108 views

Can equinumerosity by defined in monadic second-order logic?

Two properties (or concepts) $F$ and $G$ are said to be equinumerous if they have the same cardinality, i.e. if they can be put in one-to-one correspondence with each other. This can be very easily ...
3
votes
6answers
190 views

Logical issues with the weak law of large numbers and its interpretation

In several probability textbooks I have found what amounts to the following argument: Let A be an event in some probabilistic experiment. Let p=P(A) be the probability of this event occurring in ...
3
votes
2answers
144 views

Do the Kolmogorov's axioms permit speaking of frequencies of occurence in any meaningful sense?

It is frequently stated (in textbooks, on Wikipedia) that the "Law of large numbers" in mathematical probability theory is a statement about relative frequencies of occurrence of an event in a finite ...
6
votes
2answers
249 views

Set theoretic realism

What are the main contemporary arguments for and against realism about set theory?
2
votes
2answers
155 views

Is it Theoretically Impossible to Demonstrate that Set Theories Are Consistent?

I have to present on the main realist and non-realist arguments for/against set theory. According to one of my sources, it remains a matter of debate as to whether any of the set theories' (ZF, NF, ...