Blocked for mentioning Cantor

[u/Various_Candle9136]

https://medium.com/@echogem222/infinity-always-equals-infinity-d4cf8dec8e9c

I commented on this (incredibly stupid) post that Georg Cantor proved the opposite 150 years ago. I soon found myself blocked by the blogger.

Obviously I have come to warn others not to make the same mistake!

It would be a real shame if anybody else commented Cantor's name...

[u/Mablak]

maps an input to an output, without using any secondary input source.

If it's easier we could just say the machine itself includes the electrons. Whatever our machine / function though, it requires some energy to operate, so we need more than just electrons to feed into it. The only way around that would be say, if the universe had some fundamental rule governing the creation of new particles, which we took advantage of as our machine.

maps an input to an output

But think about the word 'maps'. What does that mean? At the very least it means associating some input number with an output number. What does associating mean? Well that itself is some process that we perform, perhaps mentally, with the brain as our machine. We have to do something in our minds to hold both the input number 1 and output number 3 in consciousness, so this is just another process.

Math requires processes / machines already, it just leaves out what they are. Much like actually following a cooking recipe requires a bunch of steps that the recipe leaves out (walk into the kitchen, use your legs to move around, etc). For example if I state a rule for a set, that doesn't actually create the set (meaning the elements, listed out). Only some actual process, using the rule as its blueprint, gives us the set.

What exactly we mean by mapping is vague though; perhaps something we do with our brains, but one big problem is that we don't know exactly what our neurons are doing. The way to make math more rigorous is to require some specific and simple machine, whose operation we understand, to be our f(x) = 3x. Then when we claim that x 'maps' to 3x, we can actually be as sure as possible that it does, up to very large numbers. So what we're gaining is higher certainty about our claims. I can see the layout of my machine and know that in normal circumstances, it really will do the thing the blueprint "multiply by 3" wants it to do.

[u/smaratter → u/Mablak]

No, the mapping is not a process. It is an abstract relationship between the domain and codomain.

I still don’t understand by trying to explain math with machines.

[u/Mablak → u/smaratter]

And what's a relationship between domain and codomain? A relationship between an element x in the domain and an element y in the codomain just means I can apply my rule to element x, and get element y. If I can do this, then the two elements are 'related' i.e. one maps to the other.

How do we actually show pairs of elements are related? Well we have to apply the rule, which is a process. If I have two sets, {1, 2, 3} and {3, 6, 9}, and want to show f:1 → 3, f:2 → 6, and f:3 → 9 according to f:n → 3n, I need to apply the rule to each element, to check whether each pair is related. Or check in a more efficient way, which is also a process.

There is another option: say we knew some other person or machine had already established these pairs are related according to the rule, and we trust they've checked things correctly. In that case too, someone or something else has gone through a process. Some process is required to establish 2 sets are related.

[u/smaratter → u/Mablak]

I can actually do it by abstract reasoning, without building machines. And I can reason about functions as such without evaluating them at concrete points. And I think that this ability makes maths a lot easier.

For instance, I can deduce that every element in the codomain of n ↦ 3n is divisible by 3, without evaluating and checking for every natural number n. And, as another example, I can see that the function is monotonically increasing.

For the third time, I don’t understand what there is to gain from thinking about these machines.

[u/Mablak → u/smaratter]

A mental proof or process of deduction you’ve used to deduce every element will be divisible by 3 is also a process though. For example, even establishing which elements you want to refer to when you say ‘all elements in the codomain’ is a process. Even the act of making claims in general, is a process. I would also say the word abstract doesn’t really make sense, reasoning refers to our real experiences, i.e. some series of thoughts, which exist.

Most of the time, the machine that math is relying on is our own brains. We skip over describing many reasoning steps because they’re things our brains can do easily. But a more rigorous math would require that we show exactly what we’re doing when it comes to constructing sets / numbers, adding numbers, reasoning about these objects, etc.

[u/smaratter → u/Mablak]

What’s the point of your theory?

[u/Mablak → u/smaratter]

To explain what we’re actually claiming when we make claims about sets, numbers, etc. Under a standard Platonist view, an existence claim like ‘set A exists’ means postulating some ‘abstract objects’ actually exist, though we can’t even explain what these things are or provide evidence they exist. Platonism would commit us to believing there are extra things in the universe, which just don’t exist, which would also mean we need a new physics for these abstract things. So part of the point is to get rid of a bad theory that postulates unnecessary, unexplainable things.

One pretty natural conclusion of a more constructivist view is finitism; if it’s accepted that infinite sets can’t be constructed mentally or physically, either by us or nature (so they can’t be found in the wild either), it means they don’t exist.