Three powerful arguments in support of the "random coincidence" hypothesis

[PseuTonym]

Introduction:

Consider the set of odd numbers between zero and ten. Let’s give the name u to that set.
We can write u = {1, 3, 5, 7, 9}.

Is the number 2 an element of u? In attempting to answer that question, we might pay attention to the following:

Thm(2 isn’t odd).

In other words, the statement “2 isn’t odd” is a theorem. Of course, you will not find something so trivial in Euclid’s Elements with the label “theorem.” In Euclid’s Elements, it would not even get the label “proposition.” However, we are using the word “theorem” to indicate simply that “2 isn’t odd” has been deduced from our axioms, and we don’t intend to include any connotation that the result is interesting, difficult or important.

For convenience, let us give the name P(t) to the property (t is an integer and t is odd and t is between zero and ten). When we replace the variable t with a specific value, P(t) becomes a statement.

The reason that we are interested in Thm(2 isn’t odd) is that it implies the following:

Thm(not P(2))

Question up for debate:

Is there a connection between Thm(not P(2)) and the fact that 2 isn’t an element of u?

ZF will be arguing that there is no connection.

Powerful argument number one:
ZF: “The actual connection is between (not P(2)) and the fact that 2 isn’t an element of u. That Thm(not P(2)) happens to be true is a random coincidence. Ordinary mathematical practice does not require proofs or theorems. Yes, they exist, but they are very rarely available. Ordinarily, people who are using mathematics simply make direct appeals to their mathematical intuition. Fortunately, we are all members of one species, and the human genetic endowment provides us with accurate intuition about mathematics. For example, our Pythagorean genes provide us with the intuition that tells us that the Pythagorean thing is indeed a fact.”

Powerful argument number two:
ZF: “When you say that something is a theorem, that is merely philosophical commentary about math. There can be no connection worthy of the name between the solid, pure mathematical truth that 2 isn’t an element of u, and the vague, confused, philosophical label of theorem. Ordinary mathematical practice involves direct appeals to intuition, as I have already explained. On rare occasions, somebody might prove a theorem. However, it would be a reckless practitioner of mathematics who went beyond the act of proving a theorem, and actually used the word “theorem” (or any abbreviation for that word) as part of the practice of mathematics. No, that would be commentary outside of mathematics. The first rule of Proving Theorems Club is: you do not claim that something is a theorem. The second rule of Proving Theorems Club is: you do not claim that something is a theorem.”

Powerful argument number three:
ZF: “Suppose that we use the notation u = {t: P(t)}. That notation makes reference to P(t), but it doesn’t mention Thm. Whatever is omitted from a notational abbreviation is also omitted from the thing that is being abbreviated. For example, four factorial is supposedly equal to 4 times 3 times 2, but you don’t see a factor of three anywhere in the expression 4!, do you? No, I didn’t think so. So four factorial is a number that is not divisible by 3. Nor do we see the word “not” anywhere in the notation u = {t: P(t)}. So it cannot tell us that something is not an element of u. At best it can tell us that some things are elements of u. Furthermore, I don’t see any vowels in the weird expression “Thm.” You can use the word “theorem” if you want to violate the first and second rules of the Proving Theorems Club, but you have to write all the vowels in that word.”

 

[Beryllos]

The observation that the number 2 is not in the set {1, 3, 5, 7, 9} is every bit as meaningful as the observation that the color pink is not in the set {blue, brown, white, yellow, green}.

By the way, take a look at this amazing and very useful web site:
The On-Line Encyclopedia of Integer Sequences
If you enter {1, 3, 5, 7, 9} in the search tool, it will find hundreds of different integer sequences that include the sequence 1, 3, 5, 7, 9. Some of these sequences contain even numbers as well.

 

[tonyrey]

Is the “random coincidence” hypothesis due to a random coincidence?

 

[PseuTonym]

The observation that the number 2 is not in the set {1, 3, 5, 7, 9} is every bit as meaningful as the observation that the color pink is not in the set {blue, brown, white, yellow, green}.

It is very good that you wrote that the color pink in not in your set. Somebody else might have written that the four-letter sequence “pink” is not in the set of words {“blue”, “brown”, “white”, “yellow”, “green”}. This is a reminder that although the usual decimal notation is almost completely transparent, it is nevertheless a notation. Thus, although we can visually distinguish between zero and negative zero just as we visually distinguish between 5 and -5, zero happens to be equal to negative zero. Going by visual appearance alone, somebody might deny that zero is equal to negative zero. Similarly, some people seem to think that the visual appearance of 0.999… with infinitely repeating nines indicates that it has a value less than the number one.

You introduced the question of whether or not an observation is meaningful, and that allowed you to provide your example involving colors. I appreciate your example. Is there any other contribution to this thread that can be made by focusing on the question of whether or not an observation is meaningful? Perhaps there is, but I have to admit that I don’t have a clue of how such a contribution to this thread might be made.

So, I would like to take the liberty of repeating – this time with some bolding and underlining for emphasis of a couple of words – something already posted in this thread:

ZF: “The actual connection is between (not P(2)) and the fact that 2 isn’t an element of u. That Thm(not P(2)) happens to be true is a random coincidence.”

Please note the emphasis that I have added to the words “random coincidence.” Those words also appear in the title of this thread.

Finally, I would like to repeat the question up for debate:

Is there a connection between Thm(not P(2)) and the fact that 2 isn’t an element of u?

 

[Charlemagne_III]

Is the “random coincidence” hypothesis due to a random coincidence?

:clapping:

 

[ProdglArchitect]

None of these “proofs” can actually be used to support random coincidence, because they are not based on random coincidence. You’re set of integers were selected intentionally, and therefore lose any claim to being random or to being able to discussion the nature of randomness.

As a side note, your “mathematical” proof is cute and all, but lacks any philosophical strength. It is not expressed in a way that invites discussion, nor is it expressed in a way that could be considered a logical or philosophical proof.

A better way of writing it would be:

1: There is a set of integers {1,3,5,7,9}
2: We are considering the number 2.
(From 1 & 2) 3: 2 is not a part of the given set of integers.

4: It is a random chance that 2 is not a part of the set of integers. (Note, this is where your … logic… breaks down, because it is not random chance, it is a deliberate decisions based on whatever it is you’re trying to prove.)
5: … your “proofs” which honestly don’t make any sense… Are you trying to claim that two not being in the set has nothing to do with 2 not being in the set? Because that’s what it sounds like…

 

[Beryllos]

PseuTonym, you display a marvelous exactitude with language and symbols, and a willingness to argue in a friendly way, so I am confident that you will appreciate my observation that there is indeed something odd around here, and it is u.

 

[tonyrey]

tonyrey Is the “random coincidence” hypothesis due to a random coincidence?

:clapping:Self-destructive arguments are popular in our secular society…

 

[JapaneseKappa]

“2 isn’t odd” is not a theorem, it is true by definition of “odd.”

 

[PseuTonym]

“2 isn’t odd” is not a theorem, it is true by definition of “odd.”

Are you relying upon some principle to reach the conclusion that “2 isn’t odd” is not a theorem? If you are, then please formulate your principle separately from the act of invoking the principle. If you are relying upon some principle, then I would like to be able to test the principle for myself to see how reliable it is. If you don’t formulate your principle, then I cannot test it.

I don’t know what is in your mind, but I can imagine that somebody else – somebody who agrees with what you wrote – might claim that the following principle is reliable:

“For any mathematical statement P, if P is true by definition, then P is not a theorem.”

I foresee a lot of difficulties with that principle. Of course, you did not claim that that principle is reliable. However, some lurkers who believe in that principle might be interested in the following two examples of reliability tests for that principle.

First example:
Suppose we have a definition pi = (messy expression) (See note below). Is (messy expression) = pi true by definition? If the exact definition is pi = (messy expression), then you need to use something (such as the property of equality that for all x and y, if x = y then y = x) to get to the conclusion that (messy expression) = pi.

Note re messy expression for pi:
For example, the messy expression could be a definite integral based on the area of a unit circle or the arc length from (-1,0) to (1,0) along the curve given by the function that is the non-negative part of the circle.

Second example:
(perhaps more interesting than the first example)
It requires a fairly elaborate proof to get from the axioms of set theory to the conclusion that there exists a function f such that (0,1) is an element of f, and for all positive integers n, if (n-1,y) is an element of f then (n,y*n) is an element of f. However, if you claim that what is commonly known as the “recursive definition” of the factorial function (see note below) is “true by definition”, then you can conclude that the existence of such a function f is “not a theorem.”

Note the so-called “recursive definition” of the factorial function is the following:
0! = 1, and for all positive integers n, n!=(n-1)!*n

 

[PseuTonym]

JapaneseKappa, I thought more about your question. It is thought-provoking. When you posted it, you created an opportunity for me. Perhaps I can now make better use of the opportunity that you provided.

This thread is about the general principles that allow us to answer a question of the form: “is t an element of the set u?” The thread started with an easy example of such a question. We wanted to know what general principles should be applied to answer the question “is the number 2 an element of the set of odd numbers between zero and ten?”

I raised an issue for debate, and the concept of Thm plays an important role in the debate. So we have to pay close attention to the definition of Thm. Perhaps we can resolve the problem that JapaneseKappa experienced if we revise that definition.

New definition:
Thm(P) means that (P is true by definition or P is a theorem or P is an axiom).

According to ZF, “That Thm(not P(2)) happens to be true is a random coincidence.” The ZF point of view is just one possibility. Consider another possibility …

First argument:
1-1 If Thm(not(P(2)), then 2 is not an element of u
1-2 Thm(not(P(2))
1-3 Therefore, 2 is not an element of u

After you have seen the first argument, the following might not seem new or interesting …

Second argument:
2-1 If Thm(3 is an odd number between zero and ten), then 3 is an element of u
2-2 Thm(3 is an odd number between zero and ten)
2-3 Therefore, 3 is an element of u

You could say that the second argument and the first argument are like a left hand and a right hand. They aren’t identical to each other. However, if you start with one, then you have enough ideas to reconstruct the other. You have to decide for yourself whether that makes the second argument plausible or implausible.

It may be a good idea to review JapaneseKappa’s comment now:

“2 isn’t odd” is not a theorem, it is true by definition of “odd.”

That comment brings to mind the following argument:

Third argument:
3-1. For every n, if n is divisible by 2, then n isn’t odd
3-2. 2 is divisible by 2
3-3. Therefore, 2 isn’t odd

It might be interesting to compare the third argument with the following …

Fourth argument:
4-1 For every n, if n is odd, then n isn’t divisible by 2
4-2 Therefore, 2 isn’t divisible by 2, and 2 isn’t odd

Clearly, starting from assumption 4-1, there doesn’t exist valid reasoning that will give conclusion 4-2. Let us challenge our imagination and consider what would happen if valid reasoning took us from 4-1 to 4-2. In that case, we would doubt the truth of 4-1.

An analogy will transform the assumption and conclusion of the fourth argument into a fifth argument. It won’t be a challenge to imagine that the reasoning in the fifth argument is valid reasoning.

Analogy: (n → t); (is odd → is an element of itself); (2 → Z)
(isn’t divisible by → isn’t an element of); (isn’t odd → isn’t an element of itself)

Fifth argument:
5-1 For every t, if t is an element of itself, then t isn’t an element of Z
5-2 Therefore, Z isn’t an element of Z, and Z isn’t an element of itself

The reasoning that gets us from 5-1 to 5-2 is perfectly valid reasoning. However, does there exist a set Z such that 5-1 is true? The analogy has transformed the topic from elementary number theory to set theory. The principles of logic stay the same, but we have to pay close attention to what is asserted. Valid reasoning preserves truth. In other words, if all of our assumptions are true, then a conclusion reached via valid reasoning is also true. However, an analogy does not necessarily preserve truth.

The fifth argument has a couple of things in common with the second argument. In both arguments, we prove that some specific value (of the variable t) satisfies some property or fulfills the requirements of some criterion. For example, in the second argument we prove that the value 3 is an odd number between zero and ten. In the fifth argument, we prove that Z is not an element of itself. Both the fifth and second arguments use a conditional. A conditional is a statement of the form “if Antecedent then Consequent.” For example, the second argument uses the conditional “if Thm(3 is an odd number between zero and ten), then 3 is an element of u.” The fifth argument uses the conditional “if t is an element of itself, then t isn’t an element of Z.”

The fifth argument is valid, but it violates an important paradigm. There is a paradigm for using the test of whether or not a property is satisfied to establish a conclusion about whether or not something is an element of the set associated with the property. When we conform to the paradigm, we feed the truth of the antecedent and the truth of the conditional into modus ponens (see note *** below), and we extract the consequent. For example, in the second argument we feed in the property satisfaction fact Thm(3 is an odd number between zero and ten), and we extract the consequent “3 is an element of u.” In contrast, in the fifth argument, we use the conditional alone to not only reach the consequent, but also to determine whether or not the property is satisfied. When we conform to the paradigm, we arrive at different consequents depending upon whether or not the given value (of the variable t) satisfies the property. In the fifth argument, the specific value Z (of the variable t) fails to satisfy the property, but we arrive at the consequent that we would have reached if Z had satisfied the property. The fifth argument violates the paradigm.

*** Note: Modus ponens is the rule of inference that allows us, given the conditional and the antecedent, to detach the consequent from the conditional, and to assert that the consequent is true.

 

[JamesATyler]

Now I am totally confused. Did you say that you originally posted an issue for debate? I thought you were posting an argument for refutation.

 

[JamesATyler]

I am not a mathematician and I would need to study what you have studied to speak that language more effectively but I will say what I see.

The set u = {1, 3, 5, 7, 9}. I don’t see anything in the notation that implies that this set must only ever contain odd numbers. The notation just shows a set of numbers. Looking at the set, I can determine that they are all odd, but that is just an observation. It doesn’t seem to have anything to do with the definition of u though.

Ok, the next part. Is 2 an element of u? And to start the questioning process, Thm(2 is not odd). First off, the Thm doesn’t help me determine if 2 is a part of the set since u = {1, 3, 5, 7, 9} doesn’t automatically exclude even numbers. But just to say for argument sake, that you have explicitly defined u as never containing any evenly divisible integers. If that is true, then it seems like the thm(2 is not odd) would be enough to exclude 2 from the set once the thm is proven true.

Next, I have a problem with the statement P(2). previously you say "(t is an integer and t is odd and t is between zero and ten). " So what is P(2)? 2 is an even number. If you write P(2) then you just nullified your own limitation on the property P. But for the sake of argument, let’s say we can write P(2) without nullifying the description of P.

Next. You say Thm(2 is not odd) implies Thm(not P(2)). The Thm(not P(2)) seems to be implying that 2 can not have the properties of P. Which is true by the definition you gave to P. It seems like since P(2) is not a valid statement, and that is what I think Thm(not P(2)) is saying, then Thm(not P(2)) implies Thm(2 is not odd) also.

Question up for debate:

Is there a connection between Thm(not P(2)) and the fact that 2 isn’t an element of u?

I would also argue that there is no connection between “Thm(not P(2))” and the fact that 2 isn’t in the set ‘u’.

"not P(2) would be better described as P(not 2). Because I don’t even think we can write "P(2). Or I might write P(not 2 because (thm(2 is not odd)). But just because P(t) is always an odd number, it doesn’t seem to have anything to do with u directly.

However, if this statement “Consider the set of odd numbers between zero and ten.” is binding on the set u is such a way to say that the set u only contains odd numbers then there is connection between P(t) and the set u. The connection is that they both only contain odd numbers.

I apologize if my lack of understanding of mathematical terminology is clouding my ability understand your assertions. I know what a set is and I can understand how you are using P. P is supposed to imply “not odd”, therefore P(2) should not be possible.

 

[PseuTonym]

The set u = {1, 3, 5, 7, 9}. The notation just shows a set of numbers. Looking at the set, I can determine that they are all odd, but that is just an observation.

It is just an observation, but we are dealing with just one example of a set, and it is a simple example. We might run into trouble if and when we try to generalize an observation to other sets. However, we should be able to rely upon what we can see via direct inspection or observation of one specific example. Otherwise, how could we be confident of anything in mathematics?

In the very first Proposition of Euclid’s Elements, we are given a finite, straight line segment and we construct two circles. By inspection, there exists a point where the two circles intersect. The argument presented in Euclid’s Elements fails to demonstrate that there is a point where the two circles intersect. There is a gap in Euclid’s Elements, but a remedy is available. To deny that there exists a point where the two circles meet would be to deny that the diagrams represent anything, and to imagine that no remedy is possible.

It doesn’t seem to have anything to do with the definition of u though.

Certainly u = {1, 3, 5, 7, 9} isn’t the definition of u. The definition of u is “the set of odd numbers between zero and ten.” However, I think that we should be careful about something here. Technically, it is going to be some axiom or some theorem that tells us that there exists a set of odd numbers between zero and ten. A definition cannot demonstrate the existence of that set. What a definition can do is, relying upon the existence of the set, establish a connection between the description “the set of odd numbers between zero and ten” and the name “u.”

Next, I have a problem with the statement P(2). previously you say "(t is an integer and t is odd and t is between zero and ten). " So what is P(2)?

To find out, replace t with 2 in P(t) and you get …
(2 is an integer and 2 is odd and 2 is between zero and ten).
That is a perfectly meaningful statement. The second conjunct (“2 is odd”) happens to be false, and we conclude that the statement P(2) is false. We conclude that the statement “not(P(2))” is true.

"not P(2) would be better described as P(not 2).

2 is a number. What kind of thing is (not 2)? If we begin with some statement S, then we can construct another statement T that says that it is not the case that S is true. We can write that more briefly as T if and only if (not S). If S is a statement, then (not S) is also a statement.

Thanks for participating in this thread. I will address your other comments and/or questions later. If anything above is unclear or seems incorrect to you, then please let me know.

 

[PseuTonym]

Your set of integers was selected intentionally, and therefore loses any claim to being random

The question up for debate isn’t whether or not the set of odd numbers between zero and ten is a random set.

The two positions are:

1. ZF: “The actual connection is between the falsehood of P(2) and the fact that 2 isn’t an element of u. That the falsehood of P(2) happens to be a theorem is a random coincidence.”

and

1. The alternative to ZF:
   "The connection is between Thm(not P(2)) and the fact that 2 isn’t an element of u. It is not merely a random coincidence that the falsehood of P(2) is a theorem.