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numerable diagonalization [closed]

If I understand Cantor demonstrated by using a strategy using the diagonal of a matrix, that there are decimal numbers which cannot be a part of an enumeration. We can build a Table, the rows ...
kouty's user avatar
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3 votes
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670 views

How Probable is the Philosophical Significance of Numerical Patterns in Religious Texts?

I have a Muslim friend who told me about a chapter in the Quran (the holy book of Muslims) in which he claims there is a "numerical miracle." This chapter is unique in the Quran because a ...
Rede's user avatar
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3 votes
3 answers
383 views

Why are pure powers of the empty set insufficient as a definition for ordinals?

I recently discovered a philosophical term that gives expression to a paradigm that had been circling in my head. G. E. Moore discussed the “paradox of analysis”, which is similar to what I think of ...
Julius Hamilton's user avatar
2 votes
3 answers
345 views

How small can we measure space? [closed]

I got this question after looking into transcendental numbers and I noticed how there are some distinctions that should be made from numbers and reality especially in measurement of length for example ...
How why e's user avatar
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4 votes
1 answer
111 views

Is the conceptual possibility of amorphous infinite sets "evidence against" countabilism?

Countabilism is, roughly, a family of standpoints inclusive of: There is one infinite proper set, of size ℵ0, and one infinite proper class, ℵ0ℵ0. (See about e.g. "pocket-sized" and ...
Kristian Berry's user avatar
1 vote
1 answer
38 views

Is it consistent with beginning/ending principles for an interval-based temporal logic to cover a time that is exactly ω+ω intervals long?

The SEP article on temporal logic reports on possible beginning/ending principles for instant-based temporal logics: beginning: ∃x¬∃y(y≺x) end: ∃x¬∃y(x≺y) Note that ≺ is the prior-to relation, here. ...
Kristian Berry's user avatar
2 votes
3 answers
158 views

Numbers and Time

This is my first post on philosophy stack exchange, so I apologize in advance if this question is not well-defined or if it happens to be a duplicate. If so, feel free to link the corresponding post(s)...
John Smith's user avatar
1 vote
0 answers
57 views

Infinitesimals and plural quantification

In reply to, "Does nature jump?" Mikhail Katz notes that: There is a different idea in Leibniz called the Law of Continuity. One of its formulations is the rules of the finite are found to ...
Kristian Berry's user avatar
1 vote
0 answers
90 views

Why not move from proof numbers to theories instead of theories to proof numbers?

In mathematics, they do this thing where they figure out what are called "proof-theoretic ordinals" (see this section of the SEP article on proof theory for background details) of theories, ...
Kristian Berry's user avatar
3 votes
6 answers
1k views

Why do numbers apply to such disparate concepts?

I understand numbers to be defined as objects defined to have certain convenient properties in relation to certain operations. It is very surprising that the exact same group objects should be ...
tom894's user avatar
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2 answers
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Would Frege's version of the empty set contain "parafinitesimal elements," at least from the multiversal standpoint?

Frege's definition of the empty set was not a raw extensional one: he did not simply write the partial string {} and say, "That's it: that's the empty set." His account was more intensional: ...
Kristian Berry's user avatar
1 vote
1 answer
160 views

Is there a paradox of third-order arithmetic?

Calculus, sometimes analysis or second-order arithmetic, seems more intuitive when formulated in infinitesimal terms than in terms of real-valued limits. However, the meta-theory of analysis, i.e. its ...
Kristian Berry's user avatar
3 votes
1 answer
83 views

The "slow and gradual" reduction of numbers from qualitative elements to pure quantities

In a well-known book of classical scholarship, Jaeger's Paideia, Vol.1, there is the claim that "it has been justly observed that the Greek conception of number originally contained a qualitative ...
exp8j's user avatar
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2 votes
1 answer
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Is this a legitimate way to reframe structuralism in the philosophy of mathematics?

As an umbrella term, "structuralism" has to cover realist and nonrealist versions, while also carrying through the theme of its name nontrivially (for there is a trivial way to make ...
Kristian Berry's user avatar
3 votes
5 answers
476 views

Did all numbers exist at the beginning of the universe?

So I am hoping this question spurs the thought of,"Ok, lets say they didn't all exist at the beginning of the universe then how did they all come to be?". Then the next step would hopefully ...
Joe's user avatar
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2 votes
2 answers
162 views

Discerning between a number 'x' as a Natural or Real number

The usual way of teaching is to explain the numbers that are element of the reals and naturals as being the same, this was a perfectly valid way of understanding for me, why do some consider '2' as an ...
user37577's user avatar
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0 answers
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Space as being "cardinal-like"?

I was thinking about Zermelo's critique of Cantor's reasoning for the well-ordering principle, how Zermelo characterized it as an appeal to temporal intuition, whereby time itself does the well-...
Kristian Berry's user avatar
1 vote
0 answers
41 views

Can the "doctrine of degrees of existence" be used to support the well-ordering lemma apart from the axiom of choice?

I was pleasantly surprised to read (in a Wikipedia article) that: In second order logic, however, the well-ordering theorem is strictly stronger than the axiom of choice: from the well-ordering ...
Kristian Berry's user avatar
2 votes
2 answers
175 views

Why does Plato rank numbers fundamentally below certain other ideas?

As discussed in the analogy of the divided line in The Republic (509d–511e): mathematical knowledge does not achieve the height of knowledge about ideas that are given existence by the Good itself, ...
viuser's user avatar
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4 votes
5 answers
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Are numbers, given just as mathematical objects, quantities in themselves?

If we are talking just about '5', without it being with respect to any 'amount', does the idea of the number itself as a point on a line imply that it is itself some kind of abstract 'quantity'. It ...
Confused's user avatar
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3 votes
1 answer
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Karl Marx on the meaning of 1

There is a certain passage by Karl Marx I remember in which he talks about the assumptions behind the meaning of the number 1. Marx points out that when we add together two items, such as apples (not ...
jimboweb's user avatar
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-1 votes
1 answer
116 views

Plato, the Tao, and the "first number"

My understanding of Benacerraf's identification problem breaks this problem into two subquestions. There is the format issue (the one that leads to the possible "junk theorems") and then ...
Kristian Berry's user avatar
1 vote
0 answers
111 views

Could the boundary between concrete and abstract objects be vague?

Though the SEP article on abstract objects has long weighed on my mind, I never formed much of an opinion about the question until now. The opinion I did have was negative: the ~space/~time definition ...
Kristian Berry's user avatar
-1 votes
0 answers
51 views

Mathematical "forms" as a relation of varying arity

This might be more a MathSE question, but on the other hand, it would involve a peculiar reimagining of the relation between set theory and type theory, so I'll try it out here. OK, so earlier I ...
Kristian Berry's user avatar
0 votes
3 answers
177 views

Working of Mathematical Induction

I am aware of what proof by Mathematical Induction is. I have also used it in numerous proofs. However, I don't understand formal correctness/validity of the method down to the level of Peano Axioms. ...
Ajax's user avatar
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0 votes
1 answer
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What are the philosophic positions regarding the ontology of mathematical facts?

1+1=2 and, discarding any mildly clever counter-examples that don't really matter (eg 1.4 + 1.4 = 2.8, which rounds to 3), I have a hard time imagining how the discrete quantity 1 could ever be added ...
TKoL's user avatar
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-2 votes
6 answers
282 views

On the interpretation of the number i, the imaginary

Numbers can express how much there is of a certain object. The objects can exist in the mind or in reality. You can have 1 marble, 35 marbles, and even wish for an infinite amount of them. You can ...
Deschele Schilder's user avatar
1 vote
0 answers
162 views

Do constructivists (or intuitionists) reject real numbers, except the computable ones?

SEP has a bunch of pages on what (various flavors) of intuitionists or constructivists seem to accept as a model theory or as a set theory (they actually seem to diverge on the latter, in the sense of ...
got trolled too much this week's user avatar
-2 votes
4 answers
488 views

Why is the definition of the real numbers not contradictory? [closed]

I understand that a set whose members can, in principle, be enumerated (by having a formula) can be considered as a well-defined set. Therefore, set of all even numbers, multiples of 3, and so on ...
Ajax's user avatar
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2 votes
2 answers
492 views

What does Nietzsche mean by the words ‘counterfeiting the world by means of numbers?’

In ‘Beyond good and evil’, chapter 1, section 4, Nietzsche uses the words ‘counterfeiting of the world by means of numbers’: The question is, how far an opinion is life-furthering, life- preserving, ...
Maureen Norrie's user avatar
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0 answers
53 views

What is a number? Is human's cognition sense allowed us to know numbers?

I know number is an abstract object. But I am not satisfied. Let's take number 1 represents any wholesome unit that we consider, say a goat .Then we say number 2 as 2 of that wholesome unit, here 2 ...
user42757's user avatar
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2 votes
4 answers
879 views

Why is there so little discussion / research on the philosophy of precision?

I was thinking the other day about the difference between rational and irrational numbers, and wondering whether the distinction between them is created by leaving out discussion of precision. So for ...
ControlAltDel's user avatar
0 votes
1 answer
136 views

Does the real line mean that the start and end points of any line must exist?

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line https://en.wikipedia.org/wiki/Real_number We consider the set of real numbers, ...
user avatar
1 vote
2 answers
307 views

Number, Category and Set

Can it be said that a number is a category is a set? There is such a variety of ideas on numbers, categories and sets that probably anything one says about them will be controversial, but I was ...
user avatar
0 votes
5 answers
413 views

Can anything be less than one?

Zero itself seems to be an absurd number because if there is really zero of something, then nobody has ever sensed it. But even with temperatures, we don’t really have negative and positive ...
Cannabijoy's user avatar
5 votes
1 answer
270 views

Square of Opposition with percentages?

What happens if you replace the statements of the Traditional Square of Opposition with "percentages of the subject term"? Do all the relationships from the Traditional Square of Opposition still ...
Mike's user avatar
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1 vote
7 answers
1k views

Do whole numbers other than zero actually exist?

Think about counting up: you start from 0. There are many decimals in between 0 and 1, actually, an infinite amount of decimals are there. So in the same way that there is no last number there is no ...
Brock Obama's user avatar
0 votes
1 answer
172 views

What was the "rigorous" definition of "number" for the Pythagoreans?

I am not sure if this is the right stackexchange for this question. However, I'm wondering about the following thing: We know n+ow that there are rational and irrational numbers. Pythagoras however, ...
user56834's user avatar
  • 261
6 votes
2 answers
841 views

What is the difference (if any) between the concepts of natural numbers and finite cardinals?

The definition of natural numbers from Wikipedia: In mathematics, the natural numbers are those used for counting (as in "there are six coins on the table") and ordering (as in "this is the third ...
bg5's user avatar
  • 113
4 votes
2 answers
197 views

What is the difference between concepts of number and natural number?

When reading an article about Frege on Stanford Encyclopedia of Philosophy (https://plato.stanford.edu/entries/frege/#AnaStaNum), in section 2.5 I encountered the following sentence: But though ...
bg5's user avatar
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8 votes
3 answers
3k views

How does Frege's definition of number solve the Julius Caesar problem?

How does Frege's definition of number solve the Julius Caesar problem? Frege's definition of number in the end of Foundation is such: the number belonging to the concept F is the extension of the ...
Frank Booth's user avatar
6 votes
1 answer
987 views

What are functions in the Peano axioms?

I'm posting this here because it's more of a philosophical question than a mathematical one. In set theory, we define a function as a particular type of set; and since the natural numbers are defined ...
user4894's user avatar
  • 2,967
4 votes
2 answers
140 views

How do our minds divide spaces and create "entities?"

I've been thinking about how any object can be split into infinitely smaller pieces and how we may say that there is a particular object or entity, but it has an upper portion and lower portion. In ...
Morella Almånd's user avatar
5 votes
2 answers
7k views

Why is 2+2=4 a necessary truth?

A necessary truth is something which is true in all possible worlds. How can we be sure that there is no other universe where 2+2=4 can be untrue.
stack exchange's user avatar
11 votes
5 answers
2k views

What are computable numbers, and what is their philosophical significance?

What are Computable Numbers? Is computability (or non-computability) some sort of technology-dependent characteristic of numbers (via e.g. Turing Machines)? What are the philosophical implications or ...
L.M. Student's user avatar
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8 votes
2 answers
2k views

Is number π empirical or a priori?

I used the example of π, but this applies to other transcendental numbers as well, such as e Kant classified statements into 4 epistemic categories based on two criteria: The Analytic/Synthetic ...
Alexander S King's user avatar
9 votes
2 answers
406 views

What did Poincaré mean by intuition of pure number?

To what does Poincaré refer in his article Intuition and Logic in mathematics when he speaks about the intuition of pure number? He refers also to two other forms of intuition, besides the "...
Always learning's user avatar
4 votes
4 answers
1k views

What are the "undefinable numbers" in real analysis and philosophy?

What if any important results in real analysis make use of the notion of an "undefinable" real number? (Whatever "important" may mean to the reader.) Or is it used more in the philosophy of ...
Dan Christensen's user avatar