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1answer
101 views

Where do models of set theory live?

When we are studying independence proofs we are dealing with statements of the form $Cons(T)\rightarrow Cons(T')$ where $T$ and $T'$ are first order theories; commonly $T$ and $T'$ are subtheories of ...
2
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7answers
711 views

Does it make any sense to prove $0.999\ldots=1$?

I have read this post which contains many proofs of $0.999\ldots=1$. My question is, Does it make any sense to prove this equality? Can one give any "meaning" of the symbol $0.999\ldots$ ...
16
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3answers
696 views

What did Whitehead and Russell's “Principia Mathematica” achieve?

In philosophical contexts, the Principia Mathematica is sometimes held in high regard as a demonstration of a logical system. But what did Whitehead and Russell's Principia Mathematica achieve for ...
7
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3answers
213 views

Existence of mathematical objets constructed using the axiom of choice

Let consider the Vitali set $V \subset \mathbb R$, which is constructed using the axiom of choice. (I could take any other mathematical "object" that can be constructed using the axiom of choice, but ...
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1answer
51 views

Representing a $\sigma$ - structure using a signature-$\sigma$ in Mathematical Logic.

In mathematical logic, I have a question regarding how a signature-$\sigma$ relates to a corresponding $\sigma$ structure which interprets the signature-$\sigma$ In Chiswell and Hodges book ...
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2answers
62 views

The formal definition of an interval

I is A real interval iff ∀ x,y ∈ I the segment [x,y] ⊂ I I can't understand why an interval is defined this way Why it isn't defined the same way segments are? how can the definition of an ...
0
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1answer
41 views

A box comprised of infinite number of small similar boxes.

On Wikipedia, I read, "A box can be thought of 'small boxes' infinitely repeating in all three dimensional directions" I don't understand what does Wikipedia wants to say with a box containing ...
0
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2answers
95 views

Are there alternative definitions/conceptions for the function concept?

The function is a mapping between elements of sets. However, I was thinking that would it be philosophically possible to come up with a new concept that's not a mapping between elements of sets, but ...
6
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1answer
184 views

Description of the Universe $V$ [closed]

For me, the concept "set" seams very ambiguous. This does not satisfy me because sets are used very often in mathematics, and so many questions in mathematics are not definite for me. I want to read ...
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5answers
161 views

What is exactly a “Point”? [duplicate]

I read somewhere that a line is made up of infinite points. Between any two points on that line, there are another infinite points. and between any two points BETWEEN those 2 points there are ...
2
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0answers
53 views

“Probable” truth in mathematics

This might be more of a philosophical question, but why in mathematics is the tendency to only accept formal proof as a means of finding out what's true? In the physical sciences there's no such ...
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1answer
55 views

Creating the formula and understanding them [closed]

We have, for example, the formula for velocity is distance over time ($\upsilon = s/t$). How do we get this formula? Why not ($t/s$ or $s*t$)? For this example we need physics to explain velocity, and ...
6
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2answers
63 views

What “linguistic and philosophical problems” might be inherent in trigonometry?

In "A Mathematician’s Lament", Paul Lockhart derides the "status quo" of math education, claiming that "mathematics is an art form done by human beings for pleasure" but instead is taught "devoid of ...
0
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1answer
47 views

What does Poincaré mean for intuition of pure number?

To what does Poincaré refer in this article http://www-history.mcs.st-andrews.ac.uk/Extras/Poincare_Intuition.html speaking about the intuition of pure number? My answer is that he may refer to a ...
1
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3answers
120 views

Math philosophy:about arithmetic operations and equality [closed]

This is some philosophic stuff about arithmetical operations that bothers me in some sense. During first grades, pupils are taught to perceive the arithmetical operations as "a process with a result". ...
1
vote
1answer
109 views

Can there be a lottery of the natural numbers? [duplicate]

Can there be a lottery of the natural numbers, so that every natural number is chosen equally likely? The standard answer would be "No" because: If we define a measure $\mathbf{P}$ on ...
1
vote
1answer
60 views

What sort of sentence is the Goedel Sentence (for the First Incompleteness Theorem)?

As everyone knows who have studied Hilbert's writings, he divides sentences and terms (well-formed formulas) into at least (the 'at least' in deference to Smorynski) two classes: finitary, and ideal. ...
0
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1answer
54 views

Artificial intelligence methods in mathematics

Are there aritficial intelligence methods in mathematics, automatic theorem discovery and proving? Google gives results in the opposite direction - mathematical methods of AI. Are there applications ...
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2answers
48 views

Value of a number

How does one define a value of a number? What is the value of the number 4? Asked differently, how does one show that a certain number is greater than another number? After this, one might ask how do ...
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3answers
2k views

Is there an example for an undefinable number?

This question is motivated by a comment of Robert on the question Can any Real number be typed in a computer? : Can you "think of" an undefinable number? – Robert Israel I would like to ...
5
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8answers
353 views

If a property in $\mathbb{N}$ is true up to $10^{47}$ are there reasons to think it is probably true in all $\mathbb{N}$?

You have probably heard at some point statements like that the twin prime conjecture (namely that $2$ is an infinitely ocurring prime gap) is "probably" or "almost certainly" true. Same goes for a ...
0
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0answers
39 views

How different are the positive and negative numbers?

Is there a fundamental difference between the positive and the negative numbers? Or is the difference like the one with electric charges in physics, where the other type of charge was just decided to ...
20
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3answers
511 views

Probability and measure theory

I'd like to have a correct general understanding of the importance of measure theory in probability theory. For now, it seems like mathematicians work with the notion of probability measure and prove ...
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1answer
111 views

Logic (philosophy)

Let P(x), Q(x), and R(x) be open sentences containing the variable x. Prove that if the statements for all x (P(x) implies Q(x)) and for at least x (∃x) (Q(x) implies R(x)) are true, then for all x ...
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0answers
54 views

Intuitionistic response to Russel's Paradox

I'm having a look at intuitionistic approach to mathematics, and stumbled upon a derivation of Russell's Paradox that doesn't use the LEM. (Why did mathematicians take Russell's paradox ...
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0answers
55 views

Is there any solution to Frege's criticisms of Hilbert's Geometry without the application of Model Theory?

Recently I have come across the interesting debate of Frege and Hilbert regarding the Foundations of Geometry. It seems to me that the main concern of Frege was on the Logical Consistency of Hilbert's ...
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3answers
55 views

In logic why can't “p unless q” be “q -> ~p”?

Logically, when I think about p unless q I want to say that it is equivalent to q -> ~p, but the only equivalence is ...
2
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2answers
110 views

A philosophical question about an hypothetical theorem/equation of everything

Preamble I'm not a mathematician. I'm just curious. Please forgive my pseudo formalism. Please allow me, a non mathematician, to have just questions. Definition A mathematical theorem is a statement ...
1
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1answer
78 views

Can you choose -1 as the multiplicative unit? And what is a positive number?

If one starts with the cyclic group of integers and want to introduce multiplikation the ordinare choice of multiplicative identity is the generator 1. But since 1 and its inverse -1 is sort of the ...
3
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0answers
63 views

On the (Pre-)History of Sheaf Theory

In the wikipedia page on sheaf theory I found the following statement which somehow puzzled me: some of the facets of sheaf theory can also be traced back as far as Leibniz. Could anyone explain ...
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2answers
105 views

Higher-Order Logic in ordinary Mathematics?

Do we use the language of higher-order logic in ordinary mathematics? (If yes: Can you give some examples?) Or are we always working with first-order logic? Comment: Maybe you are going to say that ...
0
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1answer
101 views

Some Philosophical Questions About Mathematics and Logic [closed]

The following questions may seem very philosophical and I guess that you guys will tell me that this is not the right place for asking them. But for me it is important to get answers from ...
0
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0answers
25 views

Completeness property in signal analysis

Why completeness is an important property for signal analysis such as Fourier? What if we don't have a such a property?Many books discuss that the vector should not have a hole to complete.what is ...
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3answers
197 views

Logic & Reality [closed]

Maybe just a quick preface first before the question. I recently started a YouTube channel where I'm trying to clear up confusions I see on various (usually philosophical topics). In my 2nd video, the ...
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0answers
57 views

Question on a note by Kreisel

In Kreisel´s "Two notes on the foundation of set theory" he writes in a footnote that 2 is to be considered as measurable and omega is also measurable. Further more he goes on to say that an ...
2
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1answer
70 views

Meaning of “existence” for an uncomputable function related to the Halting Problem

Take the set of all Turing Machines $TM$, we can divide this set in two: $P$, the set of all Turing Machines that will halt if starting from an empty tape, and $Q$, its complement: the set of all ...
0
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0answers
54 views

What is the physical significance of arithmetic operations?

Here is an example of what I mean by physical significance: When we use some geometric or trigonometric identity, let us say Pythagoras' theorem to calculate the length of the diagonal of a field, ...
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2answers
72 views

Is it possible to distinguish rest and movement in hyperbolic universe?

Imagine a large body (for example, a planet) in 3D hyperbolic space. Now imagine the planet starts moving in a straight line at constant speed. In Euclidean space, all points would move along ...
7
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4answers
800 views

As of August 2015, is the “set” of all gold medalists in the 2016 Olympics a set?

As of August 2015, is the "set" of all gold medalists in the 2016 Olympics a set? I think it is since the defining property is very clear. However, given any $x$, we do not know if $x$ is in this ...
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1answer
111 views

Proof of identity A = A or 1 = 1

Is 1 = 1 an assumption? I feel it's a very good assumption, but is there a proof for it? Imagine a world where people were contesting it, where equivalence wasn't a common sense concept. In reality no ...
6
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3answers
344 views

Plantinga's logical argument for mind-body dualism [closed]

Some may feel this is not appropriate for the mathematics stack exchange, but it is a question in logic, and I feel it is entirely a good fit. The following argument has been put forth by the ...
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votes
16answers
6k views

Why do we not have to prove definitions?

I am a beginning level math student and I read recently (in a book written by a Ph. D in Mathematical Education) that mathematical definitions do not get "proven." As in they can't be proven. Why not? ...
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2answers
105 views

About the cardinality of natural numbers [Solved]

I had learned that the set is countable if and only if it is finite or countably infinite. We know well that the set $\mathbb{N}=\{1,2,3,4,\dots\}$ is an infinite set. In order to find out if the ...
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2answers
109 views

Can math be learned backwards? [closed]

In C++, we can reverse engineer and performance binary analysis to know exactly what a piece of binary will do, even without seeing the original source code. In math, can this be done? Basically, can ...
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1answer
129 views

Math is no more valid than string theory or fan fiction? [closed]

I heard from a math expert that stated maths are not a single bit more accurate or valid than fan fiction or string theory. This was in a discussion regarding the philosophy of math and whether math ...
4
votes
1answer
138 views

Can the generalized continuum hypothesis be disguised as a principle of logic?

A cool way to formulate the axiom of choice (AC) is: AC. For all sets $X$ and $Y$ and all predicates $P : X \times Y \rightarrow \rm\{True,False\}$, we have: $$(\forall x:X)(\exists y:Y)P(x,y) ...
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5answers
278 views

What exactly is real number?

This question may sound philosophy, but it has been bothering me for a very long time, therefore I have to ask it here. The story goes back when my first time reading Apostol's Calculus, then I had ...
3
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1answer
96 views

Troubling questions about probability

Suppose we have some random phenomena. Is it true that any event concerning the phenomena has a fixed "correct" probability? That is, the correct probability is the relative number of occurrences of ...
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1answer
21 views

Finding unique rules for a finite number of initial steps, using Information theory

Is there a unique way to determine which rule provides the sequence that matches a finite number of initial steps, choosing the rule that needs the least amount of information to be described? ...
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2answers
81 views

From the perspective of the multiverse theory, would maths “work the same” in every possible Universe?

I've had an interesting discussion with a friend recently and I was arguing that in every possible Universe, mathematics would always have to work the same, i.e. $1 + 1 = 2$ would have to be true for ...