Mysticism vs. Mathematics

[alan1000]

This is a topic which I have posted in other forums, with a notable lack of significant critical response. So I try my luck here. If you reply, please do so with reasoned mathematical argument; if you begin by saying, "A set is pure consciousness of the Other..." I WILL ignore you.

A Set Theory Test Of Parity Value

For any natural number n: from a set A of cardinality n, remove one element at a time to a separate set B.

- If, at some point in this procedure, it occurs that B and A can be paired in one-one relation [ie the sets are "similar"/have the same cardinality/attain to "parity"], then n is an even number.
- If set A is exhausted and this condition has not occurred, n is odd.
- If the cardinality of A is such that no element can be removed from it [ie A is the empty set], then it will not be possible to create a second set, the question of subset parity will not arise, and n will fail to attain to either odd or even.

Assuming validity, this argument has important implications:

(1) Whereas the standard arithmetical test of "even divisibility by 2" can detect only the properties of "even" or "not even", this test reliably identifies even, odd, or nil parity status, each in its own right, and without bias towards a particular outcome.
(2) It employs no concept or procedure which is not available at the axiomatic level. In fact none of the standard arithmetical operators (+, -, x, /) need be employed in proving parity value, which is purely a problem in mathematical logic. The number 2 and the concept of "remainder" are likewise irrelevant.
(3) It might appear superficially that the test employs the operation of subtraction, which is post-axiomatic, but this is not the case; subtraction is logically tied to the relational number line, which plays no part here.
(4) We may conclude that a post-axiomatic or mathematical test which relies on any of the above can be ONLY a test, and not a proof, since the required relation of logical necessity does not exist.
(5) It is a precondition of "parity value" that the number MUST be analysable into two smaller and similar subsets or values, which can be compared for "parity" or "non-parity". This point is too often overlooked in forum discussions.
(6) Enormous logical difficulties surround the common belief that the evenness of 0 can be proved by a test of even divisibility (in one form or another). The set-theory test eliminates some of this intellectual fog by taking the position that 0 is neither odd nor even.
(7) Obviously, the proof from iterative transfer would be a cumbersome way to assess large numbers! This is the sole justification for using arithmetical shorthand techniques like "even divisibility by 2" or "remainder of 0".

[Skepdick]

This is a topic which I have posted in other forums, with a notable lack of significant critical response. So I try my luck here. If you reply, please do so with reasoned mathematical argument; if you begin by saying, "A set is pure consciousness of the Other..." I WILL ignore you.

A Set Theory Test Of Parity Value

For any natural number n: from a set A of cardinality n, remove one element at a time to a separate set B.

- If, at some point in this procedure, it occurs that B and A can be paired in one-one relation [ie the sets are "similar"/have the same cardinality/attain to "parity"], then n is an even number.
- If set A is exhausted and this condition has not occurred, n is odd.
- If the cardinality of A is such that no element can be removed from it [ie A is the empty set], then it will not be possible to create a second set, the question of subset parity will not arise, and n will fail to attain to either odd or even.

Assuming validity, this argument has important implications:

(1) Whereas the standard arithmetical test of "even divisibility by 2" can detect only the properties of "even" or "not even", this test reliably identifies even, odd, or nil parity status, each in its own right, and without bias towards a particular outcome.
(2) It employs no concept or procedure which is not available at the axiomatic level. In fact none of the standard arithmetical operators (+, -, x, /) need be employed in proving parity value, which is purely a problem in mathematical logic. The number 2 and the concept of "remainder" are likewise irrelevant.
(3) It might appear superficially that the test employs the operation of subtraction, which is post-axiomatic, but this is not the case; subtraction is logically tied to the relational number line, which plays no part here.
(4) We may conclude that a post-axiomatic or mathematical test which relies on any of the above can be ONLY a test, and not a proof, since the required relation of logical necessity does not exist.
(5) It is a precondition of "parity value" that the number MUST be analysable into two smaller and similar subsets or values, which can be compared for "parity" or "non-parity". This point is too often overlooked in forum discussions.
(6) Enormous logical difficulties surround the common belief that the evenness of 0 can be proved by a test of even divisibility (in one form or another). The set-theory test eliminates some of this intellectual fog by taking the position that 0 is neither odd nor even.
(7) Obviously, the proof from iterative transfer would be a cumbersome way to assess large numbers! This is the sole justification for using arithmetical shorthand techniques like "even divisibility by 2" or "remainder of 0".

Is there a question here, because you seem to be stuck in the truism of how the unit-measures of "evenness' and "oddness" are defined on a set?
You may not be using the standard operators (+, -, x, /) , but you are definitely using the comparrison operator (=).

And Mathematicians are sneaky fuckers. The integers are closed under (+, -, x) , but they are open under division (/). And just about everything in Mathematics is open under comparison (=, <, >)!

(1+1) = 2 (Integer)
(1/2) = 0.5 (Real)
(1=1) = True (Boolean)
(1 < 1) = False (Boolean)

Wait, what?!?! Booleans in arithmetic? Yeah... the Mystics forgot to tell you about that, didn't they?

Everything else is mechanical....

Code: Select all

import random
n = random.randint(1,1000000)
A = n * [1] # Set of size N
B = [] # Empty set

for i in range(len(A)):
	# Take an element from A and put it in B
	B.append(A.pop())
	# Compare the sizes of A and B.
	if len(B) == len(A):
		print('Even')
		break
	elif len(A) == 0:
		print('Odd')
		break

[Skepdick]

And... by the way. Here's another way to do what you are doing...

divide n by 2 and record the result.

If the result is a Real number then n is odd.
If the result is an Integer then n is even.

The below only depends on the (/) and (==) operators.

Code: Select all

import random
n = random.randint(1,1000000)
result = n/2

print(f'The number {n} is...')

if type(result) == int:
	print("Even")
elif type(result) == float:
	print("Odd")

[alan1000]

Thanks for your reply, Skepdick, but I'm not entirely sure you have understood the axiomatic aspect of the argument. The operation of division is logically reducible to an operation of iterative subtraction, which would be post-axiomatic, since it is logically tied to the relational number line (often referred to, misleadingly, as the "integer" line).

[Skepdick]

Thanks for your reply, Skepdick, but I'm not entirely sure you have understood the axiomatic aspect of the argument.

Perhaps you didn't explain it clearly enough?

The operation of division is logically reducible to an operation of iterative subtraction, which would be post-axiomatic, since it is logically tied to the relational number line (often referred to, misleadingly, as the "integer" line).

That's not true. The integers are open under division.

1/2 is not an integer.
3/2 is not an integer.

So division cannot be reducible to subtraction (with respect to the integers), because in order to divide 3 by 2 you'd have to subtract 1.5 from it e.g you have to subtract a non-integer from an integer.

[noerd]

if you begin by saying, "A set is pure consciousness of the Other..."

… if sentence’s „word“ … be’in’ … surrounded … construct !

[wtf]

This is a topic which I have posted in other forums, with a notable lack of significant critical response.

Most people would conclude that their exposition is lacking, and would work to improve it.

So I try my luck here.

You, on the other hand, choose to just post it again, hoping for a different result.

For any natural number n: from a set A of cardinality n, remove one element at a time to a separate set B.

What is the point of all this? I did read your argument on another forum. You seem to believe that 0 = 0 + 0 is not valid, and that 0 is not an even number. You're wrong. That's why your screed is getting no traction. It's because you're wrong, you're trivially wrong, and you are making a mountain not out of molehill, but out of a molehill that is not even there.

An integer is even if it can be divided by 2. Otherwise it's odd.

Or if you prefer, an integer is even if it's congruent to 0 mod 2. Same thing but the idea generalizes to arbitrary divisors, where we can partition the integers into n equivalence classes mod n.

This is not rocket surgery. It's basic arithmetic, and in the case of modular arithmetic, basic number theory.

And what, pray tell, is "nil parity status?" I'm afraid to ask. But I want to hear this.

[wtf]

And... by the way. Here's another way to do what you are doing...

divide n by 2 and record the result.

If the result is a Real number then n is odd.

Oh my. False as false can be.

Counterexample: Is 6 even? Divide it by 2 and we get 3. Is 3 a real number? Yes. Therefore 6 must be odd. Ooops!

Of course you meant, "if the result is not an integer," not "if the result is a real number." Integers are real numbers.

[Skepdick]

And... by the way. Here's another way to do what you are doing...

divide n by 2 and record the result.

If the result is a Real number then n is odd.

Oh my. False as false can be.

No, it isn't. I meant precisely what I said given the model of computation in have in mind. And it's absolutely true in that model.

Counterexample: Is 6 even? Divide it by 2 and we get 3. Is 3 a real number? Yes. Therefore 6 must be odd. Ooops!

That's not a counter-example. That's a type error.

From a type theory perspective 1:ℝ and 1:ℕ are not the same Mathematical object and 1:ℝ / 1:ℕ is a type error, not 1.
It's a damn shame your representation doesn't let you distinguish between integers and real numbers.

If you disagree - talk to my type system.

Code: Select all

utop # float(1) / int(1);;
Error: This expression has type float but an expression was expected of type
         int

Of course you meant, "if the result is not an integer," not "if the result is a real number." Integers are real numbers.

As usual you are wrong about me being wrong. Time to wrap your head around polymorphism, maybe?

What's the type signature of the division operation x/y when x and y are in ℕ?

[wtf]

No, it isn't. I meant precisely what I said given the model of computation in have in mind.

As usual, you don't know any math and falsely believe you can apply programming paradigms where they don't apply.

And what you have "in mind" is rarely what the subject of the discussion thread is about. We are not required to read your mind in order to make sense of what you write.

If you don't know that 3 is a real number, then you don't know any math. Questions of how 3 is represented in a computer program are irrelevant in this context.

[Skepdick]

If you don't know that 3 is a real number, then you don't know any math.

3 is not any kind of number, moron'. It's a symbol that can be used to represent a number.

3:ℝ represents the real number 3.
3:ℕ represents the integer 3.

Read a book. It's covered in the first chapter.

https://homotopytypetheory.org/book/

[wtf]

3 is not any kind of number, moron.

What is wrong with you? Were you raised in barn? Do you have Tourette syndrome? It's distasteful and unpleasant to engage in conversation with you. Are you like this in real life? Good luck to you.

[Skepdick]

3 is not any kind of number, moron.

What is wrong with you? Were you raised in barn? Do you have Tourette syndrome? I find it distasteful to engage in conversation with you.

Don't talk to me. Talk to the book.

"3" is not meaningful notation.

https://homotopytypetheory.org/book/

Page 18.

A good way to think about this is that in set theory, “membership” is a relation which may or may not hold between two pre-existing objects “a” and “A”, while in type theory we cannot talk about an element “a” in isolation: every element by its very nature is an element of some type, and that type is (generally speaking) uniquely determined. Thus, when we say informally “let x be a natural number”, in set theory this is shorthand for “let x be a thing and assume that x ∈ N”, whereas in type theory “let x : N” is an atomic statement: we cannot introduce a variable without specifying its type.

[Skepdick]

Are you like this in real life? Good luck to you.

I can ask the same question!

Do you randomly go and omit the type of your objects so that you can pull a sleight of hand and choose the type that suits your argument when you divide 6 by 2? What's the type-signature of the division operator?

Is 6:ℕ / 3:ℕ = 2:ℕ or 2:ℝ?


Fucking moron.

[Skepdick]

Questions of how 3 is represented in a computer program are irrelevant in this context.

Idiiot. 3 is not represented any kind of way. 3 is a representation in ALL contexts. What does it represent? 3:ℕ;? 3:ℝ? 3:Z?

https://en.wikipedia.org/wiki/Equivocation