Philosophy of Mathematics


#121

It’s not optimistic, it’s math. Perhaps that is the problem between the two of us l. I am referring to formal mathematical statements constructed in a given system, eg number theory or set theory. In any mathematical theory, the set of all possible statements is infinite, but well defined, and countable. Each statement is, by definition, either true or false. So why is it optimistic to say the set of all true statements exists?


#122

Each statement in any given theory can be mapped onto the set of natural numbers.


#123

I guess I did not really explain myself all that well.

A set of all true statements can a set of statements all of which are true, as opposed to a set in which some are true, some are false. It could have 3 or 500 true statements, all of which are true.

A set of all statements that are true would include all statements that have been made that are true, but would not contain truth that has not been stated. Newtonian physics contained all statements that were true until Einstein. Further true statements are included in relativistic physics.

The set of all possible statements that are true addresses those problems by including unstated statements and limiting them to what are true. This seems to be what is needed for the OP’s argument.

Your idea of a subset of a countably infinite set of statements may address the problem. Or not. Would a previously unstated truth belong to it? Statements that have actually been stated are finite, so I do not really follow what makes it infinite.


#124

Do you have a source for that or can you argue it more clearly yourself? It’s easy to write down such a thing.

A = {X | X is a set}

I don’t see how what you’re saying makes sense but I welcome correction.


#125

It is important if you want to develop a computer algorithm to model your theory. Many people care about that.


#126

Yes. However, geometric shapes do not exist apart from the objects whose shapes they are. Thus, circle does not exist except in things that are circular. Likewise, numbers do not exist independently of numerable things. The number three does not exist in nature as a number by itself. However, whenever there are three things in nature, the number three exists.

As stated above, mathematical objects exist in nature, but not apart from the material objects of which they are the quantitative feature. However, they also exist in the mind of the person who knows or understands them. But these are two different modes of existence. The circle may exist as an accident of a circular substance. It is a real being because the circular substance is a real substance. In the mind, however, the circle has cognitive being , rather than real being. Its mode of being is not simply being , but being known.

Humans didn’t invent mathematical concepts, but abstracted them from matter by their intellect. If you draw different circles on the blackboard, – some big, some small, some red, some yellow, and in different locations on the blackboard – and tell me they are all what you call a “circle,” then I get by abstraction what you mean by “circle.” I understand what circle means, even if I cannot give you a precise definition. I did not invent my understanding of what a circle is. I learned or discovered it.

God is the cause or creator of everything, including their mathematical properties. Everything therefore exists in the Mind of the creator. But lest you become a pantheist in thinking that everything that exists in the Mind of God is part of God’s substance when it exists outside of God, let me tell you that although things exist eternally in God’s Mind, God knows them as limited, created beings. Therefore, they don’t exist outside the Mind of God as parts of God’s Substance, but as limited creatures of God. In the world of nature material and mathematical objects exist as temporal, created real beings , although they exist eternally in the mind of God as beings known by Him from eternity. (I added the italics to show the difference in the mode of being that things have in nature and the mode of being that they have in the Mind of God. One is real existence; the other is cognitive existence.)


#127

Gosh, you don’t know this? You must be undergrad. Russell’s Paradox:
If your A exists then this subset B also exists:
B = {Y | Y is a set not containing itself as an element}
Question: Is B an element of B or not? Just try to answer… :wink:


#128

Nice to meet you again!

Some things that mathematics studies are arbitrary, but not all. The entities that mathematics deals with are not arbitrary because they were obtained by abstraction from the things of nature. There is nothing arbitrary about the associative law or the commutative law in algebra, for example. They were fixed by the nature of numbers.

However, after the fundamental concepts have been formed, other concepts can be derived from them, and arbitrary rules can be invented to see what the implications of such concepts and rules are. Regardless, the new rules still have to be consistent with the fundamental laws of mathematics that are not arbitrary.


#129

Unstated or not does not matter. That’s not a mathmatical concept. In any mathmatical theory, statements are well defined grammatically. As such, one can use the grammar rules to figure out how to “count” all statements that are feasibly possible. In other words there is a one to one (and onto for any mathmatical reading this) mapping between the natural numbers (positive integers) and each statement. Give me a number, any number, and I can return a unique statement (a grammatically correct sequence of symbols) and give me a statement, I can return it’s number. Hence, the set of all statements is countably ly infinite, but it mist assuredly exists, is it is well defined. Since every statement is true or false, the set of true statements also exists. Unfortunately, we can not determine (using only the rules of that particular theory) if any given statement is in that set.

Again, stated or unstated does not matter, it is not a mathmatical my defined concept.


#130

Yes, I already said that earlier. I said only some fields of mathematics study arbitrary constructs.


#131

Because it allows us to make statements about statements within the same theory, which provides an answer to AINg’s question.


#132

Does the Klein bottle exist or not?


#133

Your condescension isn’t appreciated. My being an undergrad is a fact I made clear in this thread, btw.

Anyhow. Thanks for the correction. I vaguely remembered such an argument, but as I couldn’t place it figured I must be mistaken. It pays to do your due diligence.


#134

They manufacture it now, don’t they? It’s expensive, though.

Are you telling me something?


#135

There are limitations on this, which I clearly do not remember all that well. I was addressing an argument that struck me as over simplified:

I still cannot place my fnger on the issue, but I think it is in the area of completeness or definition. Gödel or Tarski, I think. T cannot contain the laws that determine the truth of statements in T? Idk. I just think there is something that requires T not to include every true statement. My memory of this stuff is too rusty.


#136

The Klein bottle itself is non- self-intersecting and needs four dimensions. What they manufacture is a three dimensional representation of the Klein bottle.


#137

Just read the OP only responding to that for now but I would suggest the following:

Math is objectively true and objectively false. If we try to go beyond the limits of reasoning and say math is objective, then objectivity carries a value of true and false. It is vital that we do not say true or false or we back away from the limits of our reasoning. So objectivity is true and false a paradox.

I would suggest that paradoxes are the limits of our infinite minds. I would also suggest that we have our being or our being is limited to a finite infinity of which God is in no way limited to or by.


#138

So, by asking me whether the Klein bottle exists, you are asking me whether there is a four-dimensional (spatial dimensions) object existing in reality?

In my opinion, no. We live in a world of three spatial dimensions. An object in a four-dimensional manifold only exists in our mind as a derivative concept, that means, a concept derived from concepts of real things, but are themselves not real. In the same manner, I can say that real numbers are real, because they represent the numeration of real things. But the imaginary number i is not real. It is a derivative concept. The same is true of the Klein bottle. It is a purely mental construct.

When I said that mathematical objects exist in nature, I am referring to mathematical objects (number, lines , shapes) that are properties of real beings. These cannot exist apart from the real substances of which they are the properties. However, there are mathematical objects that are derived from other concepts; these are not always real, and may exist only in the mind of the mathematician and, of course, God.


#139

Simply not true. A mathmatical system is consistent, not complete. If it is not consistent, it becomes meaningless.


#140

We will just have to agree to disagree here


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