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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
44 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
116 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
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
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1answer
59 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. ...
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1answer
53 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 ...
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8answers
352 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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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
488 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
50 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
107 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 ...
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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
103 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
92 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 ...
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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
190 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
69 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 ...
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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
67 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 ...
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4answers
799 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
108 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
votes
3answers
338 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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16answers
5k 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
108 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
125 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
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1answer
136 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
277 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
78 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 ...
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1answer
78 views

Proving a theorem of logic

At the moment I'm going through a book which treats logic in a very rigorous axiomatic way. But I just got stuck in this theorem that I can't seem to be able to solve (I'm still trying hard). The ...
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0answers
112 views

When a function is a dimension?

The concept of dimension is used in many different contexts. Generally a dimension is a function that has as domain some family of sets ad has value on a set that, in the most common situations, is ...
5
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1answer
183 views

Since arithmetic has a model (thus it is consistent) why care if consistency can't be proved?

Since arithmetic has a model, the numbers as we know them, it is consistent. Why do we care if consistency can't be proved within arithmetic? Do I miss something, ie in what we can consider a model?
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3answers
95 views

Numbers and reality [closed]

I have a question I don't really know how to formulate, so apologies for the cloudy mess. The topic is on the meaning of numbers, that is when I say 3, I am referring to, say, $3$ avocados, or $3$ ...
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12answers
7k views

What is a negative number?

I'm trying to get to an abstract definition of a negative number that would fit in with the basic concept of addition/subtraction. There are questions here about multiplying and dividing negative ...
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5answers
1k views

What do people mean by “finite”?

Many arguments about the foundations or philosophy of mathematics centre on the question of whether or not there exist objects or entities (such as certain sets) which are not "finite". (For ...
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1answer
79 views

What exactly is wrong with this argument (Lucas-Penrose fallacy)

Argument "For every computer system, there is a sentence which is undecidable for the computer, but the human sees that it is true, therefore proving the sentence via some non-algorithmic method."
7
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1answer
155 views

Presentation of a group question

So I know that given a presentation of a group $G$, one can derive from the relations of the group presentation any element in the group $G$ right. However, I do have some confusion. If we take ...
3
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2answers
120 views

Naturality in linear algebra

Question. How can we formalize these intuitions about predicates on matrices? Let $P$ denote a predicate on matrices, so that $P(A)$ is true for some choices of matrix $A$ and false for all ...
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0answers
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Imagening the Thurston geometries

I can (more of less) imagine how it would look if space was Euclidean, spherical of hyperbolic. But there are 8 Thurston geometries see https://en.wikipedia.org/wiki/Geometrization_conjecture how ...
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2answers
378 views

Why is it important, that mathematics can be formalized in set theory?

Why is it important, that mathematics can be formalized in set theory? As one can read in the thread Are there areas of mathematics that cannot be formalized in set theory? Today known mathematical ...
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5answers
3k views

Why can't differentiability be generalized as nicely as continuity?

The question: Can we define differentiable functions between (some class of) sets, "without $\Bbb R$"* so that it Reduces to the traditional definition when desired? Has the same use in at least ...