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What exactly are the numbers we use everyday?

Pi can be defined as diameter / circunference of a circle. But what is a circle? You can't tell a computer: "build a circle and divide its diameter by its ...
2
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2answers
95 views

What is the simplest mathematical object? [closed]

What is the simplest mathematical object? I am talking about mathematics in the most abstract way possible, and not as some concrete axiomatic theory (e.g. foundational ones, like ZFC). After a lot ...
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How can I learn Math intuitively? [closed]

I am currently a Junior in High School. I am in an Intermediate Algebra class, but my teacher does not always explain things in a way I can understand. I like to learn Math intuitively, but my teacher ...
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3answers
184 views

How should a mathematically-inclined person learn descriptive statistics?

I am interested in learning descriptive statistics. But I am completely baffled, that there seem to be no mathematically rigorous books on this subject, as far as I know at least. The Wikipedia page ...
7
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3answers
173 views

Why might Dieudonne have been “begging the question” by appealing to second-order Peano Axioms?

Following a comment by Peter Smith, I've been reading A. R. D. Mathias's paper The Ignorance of Bourbaki. Parts of the paper are above my head, but I understand it well enough for my own amateurish ...
2
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2answers
39 views

Why can't a Hilbert curve be used to put the real numbers into a listable format?

There's a very good chance this question will make absolutely no sense, as my understanding of Hilbert curves is very superficial. But let me explain where my question is coming from. From my ...
4
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0answers
105 views

Can the tehniques of higher level mathematics solve most of Olympiad level math problems through straighforward applications?

Working through many Olympiad math problems(pre-undergrad) I've found that simple applications of undergrad math will solve many of them. Does this trend go on? Can it be that Putnam problems are ...
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64 views

No Proof, Just Luck

I just read about the Goldbach Conjecture and it got me thinking about probabilities. Supposing that prime numbers are somewhat randomly distributed) then if we calculate the odds of a given even ...
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42 views

Natural numbers, divisors, primes and their generalized means

Let div, nat and pri the finite sequences given in increasing order for an integer $n\geq 1$ of its divisors $1=d_1<d_2<\ldots d_{\sigma_0(n)}=n$, the first $n$ natural numbers, and the first $n$...
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0answers
48 views

How can we recognize if something is a number?

There are formal definitions of various types of numbers; natural numbers, real numbers, ordinal numbers, cardinals etc. And we all regard them as some type of number. Are there properties that are ...
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1answer
44 views

What is this ontological position called?

If one believes that certain 'abstract' mathematics-like concepts do exist, yet the mathematics we construct and develop as humans are only approximations of those real concepts, approximations shaped ...
9
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740 views

How does one refute this ultrafinitist argument?

From Wikipedia: Edward Nelson criticized the classical conception of natural numbers because of the circularity of its definition. In classical mathematics the natural numbers are defined as $0$ ...
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Can I just make this function up?

The Lambert W function was made to solve the problem $xe^x=k$ for $x$, which is given as $x=W(k)$. Could I just make a function $x=F(k)$ which solves $x\cos(x)=k$? Even though the solution has an ...
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84 views

Intuitonism and metamathematics.

There are various reasons why one would want to reject the law of the excluded middle when doing "normal" mathematics, which I won't get to here, but accepting those, does the same reasoning hold when ...
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0answers
54 views

Group Theory and Cardinals [duplicate]

I was wondering the following: Let's suppose that $G$ is a non-empty set, then, can we always find a binary operation $*$ such that $(G,*)$ is a group? For example, if we fix $G=\mathbb{Q}$ the sum ...
6
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2answers
170 views

Law of Excluded Middle Controversy

I was reading an introductory book on logic and it mentioned in passing that the Law of Excluded Middle is somewhat controversial. I looked into this and what I got was the intuistionists did not ...
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2answers
33 views

Implication as defined in mathematical logic [duplicate]

If we can take the truth of an implication to mean the validity of reasoning, then that would mean that all reasoning that begins with false premises is valid reasoning. Are there no counterexamples ...
0
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1answer
35 views

An intuitive explanation for the negatives of divergent summations?

I am looking for an intuitive explanation for why divergent summations (that are always increasing) have finite values assigned as negative. An example that is beyond "because the math says so" kind ...
3
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1answer
47 views

Prime Formulas in Heyting Arithmetic

I have been reading into Intuitionistic Logic, namely Heyting arithmetic, and I've bumped into this: Corollary 3.9. Let $A_0$ be a quantifier-free formula of $\mathscr L(HA)$. Then $$HA\vdash A_0\...
2
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2answers
69 views

Should it be allowed to apply classical logic to set theory?

It is well known, that the generalized continuum hypothesis isn't provable from the standard axiom system ZFC. GCH (generalized continuum hypothesis). For every infinite set A, there isn't a set M ...
3
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1answer
62 views

Evolution of Definitions

I try to understand how the definitions of mathematics have evolved (or formulated)... I'll use the epsilon-delta continuity definition as an example to ask my question... It may seem trivial, but it'...
47
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8answers
4k views

What does it mean to solve an equation?

This question might be more philosophical than mathematical. In school we are taught how to solve equations such as $x^2 - 1 = 0$ or $\sin(x) - 1= 0$. Solutions to these equations are quite simple. ...
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36 views

Did Ackermann produce a finitary consistency proof of second-order $PRA$?

In Wilhelm Ackermann's Doctoral Thesis (it is claimed, by Richard Zach, for one, in his paper "The Practice of Finitism: Epsilon Calculus and Consistency Proofs in Hilbert's Program", arXiv: math/...
2
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1answer
63 views

When can independence of a statement in a theory be reduced to “truth”?

Since the Goldbach conjecture is in $\Pi_1^0$, if it were proven to be independent of Peano Arithmetic, it would follow that the Goldbach conjecture is true (i.e. true in the standard model), since ...
3
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1answer
106 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 $...
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7answers
746 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 $0.999\ldots$ ...
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3answers
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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 ...
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221 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
55 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
66 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 object ...
0
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1answer
42 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
98 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 ...
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1answer
186 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
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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
56 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
56 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
64 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 ...
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1answer
52 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 ...
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3answers
129 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
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1answer
113 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 $\mathbb{N}$ ...
1
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1answer
62 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
57 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
49 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
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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 ...
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8answers
358 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 ...
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2answers
173 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 ...
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3answers
598 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
113 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 (P(...
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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 seriously?)...
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59 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 ...