∞ is a free variable

[Magnus Anderson]

Classify it then.

Sure.

[Skepdick]

Classify it then.

Sure.

Can't wait!

[wtf]

It's so deeply wired into your psychology as a Mathematician that you can't even recognize it all over your definitions.

I'm curious about how constructive mathematics handles definitions.

Suppose I define an even number as an integer that can be divided by 2 without remainder.

Then I define an odd number as an integer that's not even.

Would you have the same objection? Would you say that there are integers that are neither even nor odd?

[promethean75]

Yes man this is all above my pay grade anyway, so let me explain where i was going, I think. I don't doubt the reality or legitimacy of the concept of 'infinity' and the values given to it when it's expressed symbolically in either mathematics or computer code. In these languages, the word/value 'infinite' has a function and serves a logical and semantic purpose in a statement. It has a use.

But in philosophical language, more generally, its use is much more ambiguous. 'God is infinite', 'infinite space and time', 'infinite attributes', 'we've lived an infinite number of lives (reincarnation metaphysics)', 'an infinite line', and the like.

Consider, how would i convey to you that we've used the word 'infinite' properly to describe something like 'god' or 'space'? Against what kind of misuse could i check it? Could i be certain we had the same thing in mind when we nodded in agreement about what we meant when we said 'infinite'?

In mathematical language, on the other hand, a misuse of an assigned value like 'infinite' would produce a tangible error, and i could be certain we were in agreement about the nature of that error. There would be nothing ambiguous about it.

But the former; we may not even be in agreement about what we meant and it wouldn't matter or even disadvantage the philosophical conversation in the slightest way.

Keep infinity metaphysics free!

[wtf]

But in philosophical language, more generally, its use is much more ambiguous. 'God is infinite', 'infinite space and time', 'infinite attributes', 'we've lived an infinite number of lives (reincarnation metaphysics)', 'an infinite line', and the like.

I anticipated this point on page 1:

It's possible that you might be thinking of its casual use in popular discussions, where it has a rather vague meaning and not the precise technical meanings given to it in math.

But in math, it has the specific technical meanings I mentioned. In fact in set theory, the mathematical theory of the infinite, the lemniscate symbol is never used. Rather, there are specific symbols for various ordinal and cardinal numbers.

Skepdick is referring to the mathematical usage of infinity, not the metaphysical or philosophical meanings.

[Will Bouwman]

Given your keenness to expand our minds, are there limits to what might be constructed?

Obviously. When you lose an axiom - you lose perks with it.

Anything which implicitly depends on the full axiom of choice can't be constructed. It's a long list.

Well yes, there are limits to what you can construct with any given axiom. My question is really whether there is any limit on what you can choose as an axiom. In philosophy, the closest to any such constraint is that you can't construct an argument based on the premise that nothing exists, for the simple fact that it is self refuting. Other than that, anything goes. From what you say here:

That's all Mathematics is, child. Games with rules.

I gather you think maths is equally frivolous.

[Skepdick]

I'm curious about how constructive mathematics handles definitions.

Suppose I define an even number as an integer that can be divided by 2 without remainder.

Then I define an odd number as an integer that's not even.

Would you have the same objection? Would you say that there are integers that are neither even nor odd?

I'd ask you to define the division operator you have in mind before I answer your question.

If division by 2 is open - no integer has a remainder when divided by 2 because its type-signature is N -> (Rx{0}).
2/2 = (1.0,0)
3/2 = (1.5,0)
4/2 = (2.0,0)

Using this notion of "division" + your definition implies all integers are even. There aren't any odd ones.

If division by 2 is closed then its type signature is N-> (N x {0,1})
2/2=(1,0)
3/2=(1,1)
4/2=(2,0)

And in both cases you'd have to tell me whether 0 can be divided by 2 (with or without remainder). You can't partition nothing into two parts...

Lastly - yeah. Given the variable x bound by the quantifier x:N. x is neither odd nor even.

[Magnus Anderson]

I'd ask you to define the division operator you have in mind before I answer your question.

He's very clearly talking about integer division.

And in both cases you'd have to tell me whether 0 can be divided by 2 (with or without remainder). You can't partition nothing into two parts...

He can sidestep the entire issue by amending his definition of the "even" number the following way:

1) Zero is an even number.
2) Any other number is even if and only if it is an integer that can be divided by 2 without remainder.

Alternatively, he can define division as an inverse of multiplication, e.g. he can say that the result of "a" divided by "b" is a number "c" that when multiplied by "b" results in "a".

Finally, he can completely ditch the even/odd classification in favor of a much simpler one, e.g. happy/sad numbers. A number is said to be happy if it's greater than zero. Otherwise, it's a sad number.

The bottom line is that you're trying desperately hard to disagree.

[Skepdick]

He's very clearly talking about integer division.

Every one of those is integer divisions.

Closed Integer division WITH remainder: N / N -> NxN e.g 3/2 = (1, 1) or 4/2 = (1,0)
Open Integer division WITH remainder: N / N -> Rx{0} e.g 3/2 = (1.5, 0) or 5/4 = (1.25,0)
Closed integer division WITHOUT remainder: N / N -> N e.g 3/2 = 1 or 7/2=3
Open integer division WITHOUT remainder: N / N -> R e.g 3/2 = 1.5 or 5/4=1.25

Are you capable of grasping that requiring a remainder changes the arity of the output?
Are you capable of grasphing that a function with unary output is different to a function with binary output?

Thus integer division with remainder is a different function to integer division without remainder.

If you aren't - ask ChatGPT to explain it to you.

2) Any other number is even if and only if it is an integer that can be divided by 2 without remainder.

OK without remainder...

Closed integer division WITHOUT remainder: N / N -> N e.g 3/2 = 1.

3 is an even number then? Then again - every integer is even.

Perhaps you meant to say WITH remainder of 0?

The bottom line is that you're trying desperately hard to disagree.

I am not disagreeing with you, you are disagreeing with me.

If you don't like precision ... IDGAF.

[Skepdick]

Then I define an odd number as an integer that's not even.

The semantics aside - this step is not constructively valid.

Negation is involutive in a classical setting because Excluded Middle. It's not involutive in a constructive setting.

For this particular instance you need to prove that odd and even are fully exhaustive categories first before you can define an involutive relaton between the "odd" and "even" predicates.

[Magnus Anderson]

Every one of those is integer divisions.

They aren't. Your "open division" is not an integer division because the quotient is not an integer.

Thus integer division with remainder is a different function to integer division without remainder.

The output of an integer division is a pair consisting of the quotient and the remainder.

If you aren't - ask ChatGPT to explain it to you.

Perhaps you should listen to your own advice. Go ahead and ask ChatGPT what it thinks of your nonsense. Or do you want me to do it instead of you and share the embarrassing results with everyone here?

2) Any other number is even if and only if it is an integer that can be divided by 2 without remainder.

OK without remainder...

Closed integer division WITHOUT remainder: N / N -> N e.g 3/2 = 1.

3 is an even number then? Then again - every integer is even.

Perhaps you meant to say WITH remainder of 0?

That's what "without remainder" means, dummy. It means "with remainder that is equal to 0".

[Skepdick]

They aren't. Your "open division" is not an integer division because the quotient is not an integer.

I am dividing two integers. That's integer division.

There are multiple possible ways to represent the result of integer division.

The output of an integer division is a pair consisting of the quotient and the remainder.

That's one possible output of integer division - yes. N / N -> N x N

Another possible output of integer division is the quotent. WITHOUT REMAINDER: N / N -> N

Perhaps you should listen to your own advice. Go ahead and ask ChatGPT what it thinks of your nonsense. Or do you want me to do it instead of you and share the embarrassing results with everyone here?

The only embarrasment here is you.

That's what "without remainder" means, dummy. It means "with remainder that is equal to 0".

WITHOUT remainder means WITHOUT REMAINDER dummy:3/2 = 1 WITHOUT REMAINDER.
WITH remainder means WITH REMAINDER dummy: 3/2 = 1 WITH REMAINDER 1

integer division WITH remainder: N / N -> N x N
Integer division WITHOUT remainder: N / N -> N

One is unary one is binary. Do you understand the difference between 1 and 2?

❯ ipython
In [1]: 3 // 2
Out[1]: 1

In [2]: divmod(3,2)
Out[2]: (1, 1)

[Magnus Anderson]

I am dividing two integers. That's integer division.

No. That's a division where the dividend and the divisor are integers. Integer division "refers to the process of dividing two integers and obtaining the quotient without any fractional part". The quoted part is from Science Direct. Google it. Or ask ChatGPT if you prefer it over Google.

That's one possible output of integer division - yes. N / N -> N x N

Another possible output of integer division is the quotent. Without the remainder.

Every integer division has a remainder even if that remainder is a zero.

You're getting seriously distracted.

The only embarrasment here is you.

Delusional beyond belief.

WITHOUT remainder means WITHOUT REMAINDER dummy:3/2 = 1 WITHOUT REMAINDER.
WITH remainder means WITH REMAINDER dummy: 3/2 = 1 WITH REMAINDER 1

One is unary one is binary. Do you understand the difference between 1 and 2?

You're SERIOUSLY getting distracted, dumbass. Every integer division has a remainder regardless of the arity of its output. You're focusing on completely irrelevant stuff in an effort to appear smart when in reality you're actually extremely dumb ( evident by the fact that you can't tell how irrelevant your focus on the arity of the output is. )

[Skepdick]

No. That's a division where the dividend and the divisor are integers. Integer division "refers to the process of dividing two integers and obtaining the quotient without any fractional part". The quoted part is from Science Direct. Google it. Or ask ChatGPT if you prefer it over Google.

Great! So you you agree that when using integer division 3/2 = 1 without any fractional part ?!?

Every integer division has a remainder even if that remainder is a zero.

Contradiction.

https://mathworld.wolfram.com/IntegerDivision.html

Integer division is division in which the fractional part (remainder) is discarded

You're getting seriously distracted.

Not really. I'm just dismissing you.

Delusional beyond belief.

Not really. I'm just dismissing you.

You're SERIOUSLY getting distracted, dumbass.

Not really. I'm just dismissing you.

Every integer division has a remainder regardless of the arity of its output.

Sure. And you are too dumb to distinguish between operators which return the remainder and operators that don't.

integer division WITH remainder: N / N -> N x N
Integer division WITHOUT remainder: N / N -> N

An operator which returns the remainder has arity 2.
An operator which doesn't return the remainder has arity 1.

You're focusing on completely irrelevant stuff in an effort to appear smart when in reality you're actually extremely dumb ( evident by the fact that you can't tell how irrelevant your focus on the arity of the output is. )

I am never the dumbest person in any room you are in.

Do you understand English? In what lagnuage do you need me to spell it out for you? Take your conceptual normatives and shove them up your ass.

[Skepdick]

He's very clearly talking about integer division.

Integer division "refers to the process of dividing two integers and obtaining the quotient without any fractional part

I swear - I can't make this shit up....

If he is "clearly talking about integer division" and integer division discards the remainder how the hell are you going to discern an odd number from an even one?!?