So the rule is that for every number, one can add one. The rule only generates one new number. One has to see the rule in a different way in order to understand infinity: imagine a number bigger than any number the rule could generate.. — Banno
Anther way to approach it that the rule "For every number, you can add one. to make a bigger number" is not generating all the numbers, but only the integers. We can find infinity by calculating 1 divided by 3 — Banno
Sure, but we were talking about counting, not pure maths. The contested statement was:
Counting infinity has nothing to do with time. — Metaphysician Undercover
One can count an infinity conceptually, without time. — EnPassant
That's not counting though. Anyone can make up a new definition of "counting", and use that definition to make whatever conclusion one wants to make about infinity. But that conclusion would be irrelevant to what "counting" really means to the rest of us. So if Cantor turned "counting" into some sort of abstract concept which has nothing do with the act of counting, as we know it, I don't see how that's relevant. You are just arguing through equivocation. — Metaphysician Undercover
So how does a constructionist handle such a number? Do they deny that the set of all numbers is properly mathematical? — Marchesk
The difference is really semantic. Counting is about associating a number with an object; 1 orange, 2 apples etc. But Cantor counts numbers with numbers by associating numbers with other numbers. In this way Cantor associates/counts the rational numbers with integers and comes to the conclusion that there are enough integers to count the rationals. — EnPassant
The question here is What does 'real' mean when we are talking about (what seem to be) abstractions? What does 'exist' mean in the context of numbers existing? — EnPassant
Setting side those never ending debates, what does it mean for a constructionist to be able to offer a proof for any conjecture involving an infinite sequence, such as any number greater than two is the sum of two primes? — Marchesk
But this comes back to what we mean by 'exist' in relation to numbers in a Platonic sense. What does 'exist' mean? — EnPassant
In mathematics infinity is a set, such as Aleph Null, not a process. Infinity is not 'the biggest number' it is all numbers, together. — EnPassant
Here is a thought. Write the squares of numbers like this-I don't know, but it's difficult to say that math is entirely made-up when it's so useful in scientific theories. Quantities of things exist, so does topography and function. — Marchesk
More or less true in set theory, a particular branch of mathematics. My area was complex analysis and when I deal with the concept of infinity it is in the sense of unboundedness of sequences or processes. — jgill
And the consequence of that is that talk of extension in mathematics becomes fraught with ambiguity. Hence, Wittgenstein's argument that mathematical extensions must be finite, and hence his adoption of finitism, seems misguided. — Banno
A number line is an irrational conflation . . . that's why the idea creates so many problems — Metaphysician Undercover
And the consequence of that is that talk of extension in mathematics becomes fraught with ambiguity. Hence, Wittgenstein's argument that mathematical extensions must be finite, and hence his adoption of finitism, seems misguided. — Banno
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