## "1" does not refer to anything.

• 3.6k
But can be thought of as correlating with linear time, each step separated from the next by a short period of time.

Wouldn't that require time to be discrete?
• 530
So in what sense does it mean to say that 1,2,3... goes on ad infinitum?

Infinity just means 'without end'.
• 3.6k
But there's no such thing as a constructed sequence that doesn't end.
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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..

We can do that, but does that work for construction? You're saying imagine a number bigger than any number the rule can construct.
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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

Sure, but then you have the problem of how the .333 repeats forever. It can't already exist on the pain of Platonism, nor can it be generated by a rule.

It seems like you're having to step outside the rule to add something. And what is that? The idea of the rule repeating forever.

So then "infinity" means a rule that never ends, but can't be generated.
• 335
Sure, but we were talking about counting, not pure maths. The contested statement was:

Counting infinity has nothing to do with time.

Cantor uses Aleph Null to count infinities. One can count an infinity conceptually, without time. How much time is there between the digits of pi? Likewise with the empty question 'What came before the beginning of time?' The real question is "What gives rise to time?" or "On what necessary condition is the world/universe contingent?" It is really an ontological question.
• 335
Infinity just means 'without end'.

In mathematics infinity is a set, such as Aleph Null, not a process. Infinity is not 'the biggest number' it is all numbers, together.
• 7.2k
One can count an infinity conceptually, without time.

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.
• 335
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.

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.
• 3.6k
In mathematics infinity is a set, such as Aleph Null, not a process. Infinity is not 'the biggest number' it is all numbers, together.

So how does a constructionist handle such a number? Do they deny that the set of all numbers is properly mathematical?
• 335
So how does a constructionist handle such a number? Do they deny that the set of all numbers is properly mathematical?

Kummer, Cantor's arch enemy, was a kind of constructionist and denied the reality of real numbers. I guess they just don't agree. 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?
• 7.2k
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.

Yes the problem is semantic, that's what I said. If you give "counting" whatever meaning you want, you can do whatever you want with it. So Cantor is not really counting as "counting" is commonly used, because number is a concept, and not a thing which can be counted. It's only by making numbers into mathematical objects (Platonism) that Cantor can count numbers.
• 3.6k
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?

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?
• 335
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?

It seems to me that a pure constructionist cannot even admit that there are an infinity of natural numbers. Induction proves that there are and many mathematical proofs rely on induction. But this comes back to what we mean by 'exist' in relation to numbers in a Platonic sense. What does 'exist' mean?

One unresolved question in philosophy is why there is something rather than nothing. We don't know but we know there is something. This necessary something that is, before all created things, is what is, eternally. This eternal substance is existence. It is not that this necessary something has the property 'existence' it is existence because existence cannot be a property. So, if numbers exist, they must be intrinsic to existence. And since it takes Mind for numbers to exist, existence must be Mind, if numbers are in existence. The only eternal mind in which numbers can exist is God's Mind.

What all this means is that existence, mind, and God are three names for the same thing.
In this context I am using the word 'existence' to mean that which necessarily is.
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But this comes back to what we mean by 'exist' in relation to numbers in a Platonic sense. What does 'exist' mean?

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.
• 637
In mathematics infinity is a set, such as Aleph Null, not a process. Infinity is not 'the biggest number' it is all numbers, together.

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. :cool:
• 335
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.
Here is a thought. Write the squares of numbers like this-

1 squared = 1
2 squared = 4
3 squared = 9 etc.

Now, you can plot this sequence of squares on a graph as a quadratic curve, the curve of x^2.

The question is, how can a flat piece of paper receive this concept of squared numbers so faithfully? How is it that it is possible to translate a thought about numbers onto a graph in flat space?

This can only be possible if there is a natural correspondence between mind and space. If mind and space were utterly different it would not be possible to create an image of mathematical ideas on a flat space. But if there is a natural correspondence between mind and space what is it? The only common factor I can think of is mathematics. That is, mind and space must be intrinsically mathematical.

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.

Yes, but the limit can be defined independently of time.
• 637
Yes, but the limit can be defined independently of time.

Of course it can. I merely mentioned a kind of isometry between iteration and time. I deal with infinite sequences whenever I dabble with research, and I rarely consider a correlation with the passage of time.
• 8.4k
it can't be said to go on forever.

Forever...

Not all forevers are temporal. A line does not require time.
Counting is a temporal process.
Have you never seen a number line? • 7.2k

A number line is an irrational conflation of two incompatible terms, discrete numbers, and a continuous line, that's why the idea creates so many problems. If there is a line which extends "forever" it is spatial. If the number line is supposed to count forever, then "forever" is temporal. But the simple line which extends for ever doesn't count anything.
• 8.4k

So this sort of thing must be a real bitch for you... • 7.2k

It's quite boring, if that's what you mean.
• 3.6k
So you believe the infinite number line exists? What happened to construction?
• 8.4k
What is this a picture of?
• 8.4k
So you believe the infinite number line exists?
exist?
• 3.6k
exist?

Have we left the question of Wittty's finitism behind?
• 8.4k
https://thephilosophyforum.com/discussion/comment/405811

and

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.
• 637
A number line is an irrational conflation . . . that's why the idea creates so many problems

It makes doing mathematics like walking across a minefield! :fear:
• 335
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.

Here's a thought. Draw the X axis. The segment between 0 and 1 is a physical extension in space. This segment contains an infinity of dimensionless points: ie points of zero dimension. But if you set down an infinity of these zeros side by side, you get 1 unit of length. The implication is that 0 x $\infty$ = 1.
• 357

So essentially any number would not refer to anything either? If so what does zero refer to? What differentiates 1 from 0?
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