12
votes
Accepted
What are computable numbers, and what is their philosophical significance?
All mathematical formalizations of (intuitive) computability are known to be equivalent, in particular they are all equivalent to computability on the universal Turing machine. So technological ...
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10
votes
Why is there so little discussion / research on the philosophy of precision?
It think there's something confused in this question. Mathematical constants like pi are exact. Pi is the ratio between the circumference and diameter of a perfect circle on a flat plane, and the fact ...
- 18.4k
9
votes
Accepted
How does Frege's definition of number solve the Julius Caesar problem?
Here is some historical context. In Grundlagen der Arithmetik (1884) Frege introduced his ill-fated Axiom V, now known as the axiom of unrestricted comprehension: every predicate defines a class of ...
- 42.5k
7
votes
Accepted
What are the "undefinable numbers" in real analysis and philosophy?
The notion is important in mathematical logic and model theory, but not in classical mathematics, including real analysis as traditionally understood. Definable predicates are generally important in ...
- 42.5k
7
votes
Accepted
What is the difference (if any) between the concepts of natural numbers and finite cardinals?
It is fair to say that the concepts are "equivalent" in the modern mathematical practice. However, they have different histories and different overtones of meaning. The notion of natural or counting ...
- 42.5k
6
votes
Accepted
What are functions in the Peano axioms?
An Intuitive Walkthrough of PA as a formal system
*Peano Arithmetic are a set of axioms in first order logic that describe how arithmetic of the natural numbers works. A first order formal language is ...
- 2,841
6
votes
On the interpretation of the number i, the imaginary
Numbers are used for more than just counting things. You are thinking of cardinal numbers, which represent the size of a set; in your case, if you have 5 marbles then your set of marbles has ...
- 718
5
votes
Accepted
What is the difference between concepts of number and natural number?
The point raised in the quote is not the same as the question that you are asking.
In the quote: It is a difference whether we define what one is, and then we define what two is, and so on, or ...
- 935
5
votes
Why is 2+2=4 a necessary truth?
The issue is complex and any "significant" answer is hardly reducible to the Yes/No pattern.
In modern mathematics, 2+2=4 is a theorem of arithmetic provable from Peano axioms.
In a nutshell, ...
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5
votes
Accepted
Can anything be less than one?
Your question is paralleled by the reaction of the Roman world to Indian ('Arabic') numerals. Accountancy was done in Roman numerals until the 1800s, exactly because of suspicions about the 'realness' ...
- 20.5k
5
votes
How does Frege's definition of number solve the Julius Caesar problem?
In a nutshell, the issue arises from the definition of number of as a second-order concept (i.e. a numerical quantifier) in Die Grundlagen der Arithmetik (1884).
Consider e.g. 0xϕ(x)=df Card[xy] (y ≠ ...
- 36.1k
5
votes
Accepted
Why is there so little discussion / research on the philosophy of precision?
You can address this issue through significant figures, which indicate the level of precision with which a value has been measured. If I say that I walked 2 miles, that's a rough estimate - I'd be ...
- 1,177
5
votes
Accepted
Karl Marx on the meaning of 1
This strikes me as something from early in "Das Kapital". I skimmed the first four chapters without finding an exact match. The closest I found was this:
Let us take two commodities, e.g., ...
- 18.4k
5
votes
Why do numbers apply to such disparate concepts?
You should not be surprised. Numbers were conceived to model concepts such as length, area, angles and so on. Any qualities that vary by extent can be compared in an analogous way. For example, the ...
- 12.9k
4
votes
Accepted
Number, Category and Set
For category theory, see Natural number object.
For set theory, see Set-theoretic definition of natural numbers.
And see Category of sets for a link between the two theories.
See also the post ...
- 36.1k
4
votes
Is number π empirical or a priori?
Kant only had three epistemic categories, analytic a posteriori are highly problematic (even Kripke talks only about necessary a posteriori). As for π it was originally defined as a ratio of the ...
- 42.5k
4
votes
Accepted
What did Poincaré mean by intuition of pure number?
"Intuition of pure number" is the intuition Poincare inherited from Kant's a priori form of perception in time. Kant recognized two forms of perception that produce synthetic a priori, and therefore "...
- 42.5k
3
votes
What are computable numbers, and what is their philosophical significance?
The computable numbers are not technology dependent. A universal computer can simulate any finite physical system to any desired degree of accuracy. And it can simulate not just the input and the ...
- 7,420
3
votes
What are the "undefinable numbers" in real analysis and philosophy?
Running off the idea that there are undefinable numbers because there are countably infinite definable real numbers and uncountably infinite real numbers, one result that could be considered "...
- 17.5k
3
votes
Why is the definition of the real numbers not contradictory?
This is a misconception, you don't need to be able to enumerate the elements of a set.
In naive set theory (which has its problems but is useful to explain the set concept here), a set is defined by ...
- 2,232
3
votes
Why is the definition of the real numbers not contradictory?
Most mathematicians are happy to use ZFC set theory or one of it's equivalents. These set theories support the "normal" real numbers.
There are, however, mathematicians such as the ...
3
votes
On the interpretation of the number i, the imaginary
In electronics, we use imaginary numbers to represent the reactance of certain components (capacitors and inductors). The overall impedance of a circuit is then a combination of the various components'...
- 2,522
3
votes
What are the philosophic positions regarding the ontology of mathematical facts?
I have my Bachelors' degree in Math, so I can tell you what the pros think about it:
https://aimath.org/textbooks/approved-textbooks/judson/ is an example of a college-level text covering the subfield ...
- 131
3
votes
Working of Mathematical Induction
The Peano axioms are dependent on first-order logic. In first-order logic, we already have the ability to make statements such as "for all n, P(n) implies P(n+1)" - except that "n+1&...
- 1,733
3
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Working of Mathematical Induction
It doesn't matter if "n+1" works as we understand it. It doesn't matter what the topic is or what the successor operator is, so long as it follows the axioms. For example, suppose I lay out ...
- 7,444
3
votes
Accepted
Are numbers, given just as mathematical objects, quantities in themselves?
They are abstractions, ways of grouping and pointing at what things can have in common. When we use mathematics on the world, we make choices about formal equivalence, eg of one orange with another, ...
- 20.5k
3
votes
Did all numbers exist at the beginning of the universe?
Did all numbers exist at the beginning of the universe?
The sensible answer seems to be that they didn't but we may need to explain why.
First, inevitably, we have to say what we assume a number to ...
- 6,221
2
votes
What are computable numbers, and what is their philosophical significance?
Most simply, computable numbers -- including computable problems -- are capable of being solved by a computing device such as calculators, computers, and, well, humans if you are so inclined ...
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