Philosophy of Mathematics

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curious_cath said:
How do you formalize the world/ the creation? The topic, if I understand correctly, began with a very ambitious approach about thinking of God as a formal system that includes mathematics.
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I didn’t say you could. I simply stated (in the context of mathematics) that the set of true statements exited (if I remember how my participation in this thread started).
 
I don’t know, its been so long since I studied all of this language structure stuff, I don’t remember all the details, but I am pretty dang sure that if there is an upper bound on sentence length, it is certainly not uncountable (sorry for the doublt negative)
 
Socrates92 said:
Mathematical truth is eternally true
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Can you give some further explanation, because im not sure what you are asking.

To me God is truth, And if anything else exists it is true because of God, and not a self-existing truth. In other-words it is contingently true. Without God, there is no truth because nothing would exist, which is, i think, impossible.

If there was absolutely nothing, it cannot be true that 2 + 2 = 4, because it is only truth because of the nature of that which eternally exists. It cannot be true because of nothing, because there is absolutely nothing in nothing, and something that is necessarily true cannot begin to be true because it is necessarily true. So there must be a reality that necessarily exists.
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curious_cath said:
The topic, if I understand correctly, began with a very ambitious approach about thinking of God as a formal system that includes mathematics.
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That’s a misunderstanding of my OP. “Math is part of God” is not the same as “math can explain God”. Which is what you’ve suggested I said.
 
chessnerd321 said:
“Math is part of God”
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A problem you are going to encounter right off the bat is whether God has parts. According to Aquinas, no. God is a simple being. Nothing can be a part of God. ( (Summa theologiae I, q. 3).
 
I’m using informal speech to describe my views. If you want a full idea of what I mean you’ll have to follow the post. Most posters here have been a part of this discussion for a couple weeks now.

Sure God doesn’t have parts, but God cant be boiled down to one word descriptions either. God is love. God is Mercy. God is Truth. God is not only love or only Mercy or only Truth.

So God is math. Doesn’t mean “God is described by math” or “God is only math”. “God is not only math” is what I mean when I say “Math is part of God.”
 
chessnerd321 said:
I’m using informal speech to describe my views.
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Just making sure that you are not confused. When discussion theology on a Catholic website, it’s best not to use terms loosely.

How much philosophy have you studied? If you don’t have at least the presocratics, Plato, Aristotle and Aquinas under your belt, your basically spinning your wheels dabbling in theological speculation. I’d also toss in Descartes and Leibniz, as well.

There is a vast body of literature out there on the philosophy of mathematics and its intersection with theology.

A good place to start is the following book:

Hacking, Ian. Why is there philosophy of mathematics at all? Cambridge University Press, Cambridge, 2014.

After that, read:

David Corfield’s Toward a Philosophy of Real Mathematics

Those will give you a good overview of the field. The second one is especially good, because it is written by a mathematician rather than by a historian of philosophy who doesn’t really know much about what math is.

Good luck!
 
It does seem you are right, if a language has M words, and the maximum length of any statements is bounded by N, it does seem like the number of statements has to be less than or equal to M^N, which although is very, very large number, is certainly finite. But, I cannot help recalling that the number of statements in a formal math theory such is countably infinite. Furthermore, I recall from studying natural language processing that if the length of a sentence is not bounded (grammatically it is not, in the real world it is), then the number of sentences is uncountably infinite. So something doesn’t make sense. I would have to go back and do some research. And I don’t have time to do so.
 
tafan2 said:
I cannot help recalling that the number of statements in a formal math theory such is countably infinite.
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That is the conclusion I reached. For setnences of length N, the number of possible statements is s < M^N. That means the sentences of all lengths are a countable infinity.

Natural language processing is a whole nother world, of which I am completely ignorant. I am guessing our creation of upper bounds is a problem, since smaller units like letters and words are not single meaning units. But that is just a guess. I was too busy walking miles to school through 3’ of snow to fill my head with such ideas.
 
rom said:
Can you show how a 4-dimensional sphere is real?
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No. It cannot be shown, because we look in 3 dimensions.

If you have a measure of time that is equal to a measure of length, then the 4 dimensional sphere will be all those points equidistant from a single point origin. We don’t really have a way to compare time and length, and the calculation for the sphere should probably be the imaginary x2+y2+z2-t2=1, so We do not see how to visualize it.

But they exist, despite our limitatations.
 
Dovekin said:
No. It cannot be shown, because we look in 3 dimensions.

If you have a measure of time that is equal to a measure of length, then the 4 dimensional sphere will be all those points equidistant from a single point origin. We don’t really have a way to compare time and length, and the calculation for the sphere should probably be the imaginary x2+y2+z2-t2=1, so We do not see how to visualize it.

But they exist, despite our limitatations.
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By “show me” I am not asking you to help me visualize it. I know it’s hard to visualize. I am asking if you could show me that it is real, that means, that it exists outside our minds.

You gave me a definition of a 4-dimensional sphere, x^2 + y^2 + z^2 + t^2 = 1, but that does not prove that the 4D sphere exists outside our mind. Definitions don’t make things exist outside the mind. I can define a mermaid as a creature half-woman, half-fish. But that does not mean a mermaid exists outside my mind, does it? I can even define it as an existing thing. But that will not make it exist in reality either. It will only mean that it exists in my mind.

So, how are you so certain that a 4-dimensional sphere exists?
 
The given 4th dimensional sphere formula given adequately describes the propagating wavefront of an electromagnetic wave. That can be tested and measured.
 
StephieNorthCo said:
The given 4th dimensional sphere formula given adequately describes the propagating wavefront of an electromagnetic wave. That can be tested and measured.
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Granted. Does that mean the electromagnetic wave is a 4D sphere and that, therefore, the 4D sphere is real? Or does it merely mean that it is OK to think of the electromagnetic wave as a 4D sphere for the purpose of mathematically predicting its propagation into space? It is a mathematical model, useful perhaps, but it is only in the mind. I have known electrical engineers who used a spring-mass system to set up the differential equations that describe the behavior of alternating current circuits. Would you therefore equate an alternating electric circuit to a spring-mass system?
 
tafan2 said:
It does seem you are right, if a language has M words, and the maximum length of any statements is bounded by N, it does seem like the number of statements has to be less than or equal to M^N, which although is very, very large number, is certainly finite. But, I cannot help recalling that the number of statements in a formal math theory such is countably infinite. Furthermore, I recall from studying natural language processing that if the length of a sentence is not bounded (grammatically it is not, in the real world it is), then the number of sentences is uncountably infinite. So something doesn’t make sense. I would have to go back and do some research. And I don’t have time to do so.
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There is a difference between saying “all sentences are of finite length” and “the length of all sentences is bound by some number N.” You can have an unending list of increasingly larger numbers. Each number on that list is a finite number, but there is no number N that is greater than all of them.

To say “all sentences are of finite length” means we are not going to consider any sentence that is of infinite length.
 
I am not sure what you are asking. Do you accept that every point can be described by 4 coordinates (x,y,z,t)? If so, then the points equidistant from an origin point would exist.

There are a number of problems that might make this imaginary rather than real:
  1. The measure of t might not equal the measure of x. We measure length and height by inches, time by seconds.
  2. Physicists use x2+y2+z2-t2=1 instead of adding all four. That introduces an imaginary element, ti. That might not fit your definition of real.
  3. Only the present is real. If this is your viewpoint, than objects that span time, as a 4d sphere must, exist only as a 3d artifact in 3d space…
These do not seem insurmountable to me, ie we can figure some way to alter definitions to allow a 4 d sphere to exist. Otherwise, how can a 4d sphere, that is the set of all points equidistant from an origin in 8 direction, not exist? What definition of reality makes you doubt it?
 
LeafByNiggle said:
To say “all sentences are of finite length” means we are not going to consider any sentence that is of infinite length.
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I agree, that is what the phrase means. I think that is by the nature of mathematical statements, but I do not know. I showed that the set of all finite length statements is a countable infinity. I have no idea how infinite length statements would affect that, but it probably makes the set of statements uncountable. IDK.
 
Dovekin said:
Do you accept that every point can be described by 4 coordinates (x,y,z,t)?
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In a 4D world, yes.
Dovekin said:
If so, then the points equidistant from an origin point would exist.
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Yes, it would exist in a 4D world. But does the 4D world exist? That’s the question.
Dovekin said:
There are a number of problems that might make this imaginary rather than real:
  1. The measure of t might not equal the measure of x. We measure length and height by inches, time by seconds.
  2. Physicists use x2+y2+z2-t2=1 instead of adding all four. That introduces an imaginary element, ti. That might not fit your definition of real.
  3. Only the present is real. If this is your viewpoint, than objects that span time, as a 4d sphere must, exist only as a 3d artifact in 3d space…
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All three points you mentioned are correct. I am happy you mentioned them because that means you understand the issues. They are the reasons why a 4D sphere cannot exist in the 3D world outside our mind.
Dovekin said:
These do not seem insurmountable to me, ie we can figure some way to alter definitions to allow a 4 d sphere to exist.
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I don’t know what definitions you can propose to make a 4D sphere exist. Definitions don’t make things exist except in the mind. To make anything exist outside the mind, then you need the power to create being, and only God, who is Being itself, can do that.

Of course, if you redefine reality itself to mean anything the mind can think of, then yes, you can make a 4D, 5D, 6D… sphere real. But I am going by the common understanding of reality as the world of beings existing outside our mind, and this consists of (a) the material world in which we live, and (b) the spiritual world (God, angels, souls). The material world is a 3D world; the spiritual world is a world with no defined dimensions. A 4D sphere cannot exist in a 3D world because it has one dimension too many. On the other hand, it cannot exist in the spiritual world because any dimension in the spiritual world is meaningless. Now, if we assume that between the material world and the world of spirits, there is a world of beings existing in a multi-dimensional world of D>3, then we enter the world of science fiction. You can’t make them real just by defining them to be real.
 
rom said:
My definition of truth is the correspondence of the mind with reality.
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I wouldn’t agree with that since reality can be chaotic and fuzzy. If you are talking about the correspondence theory of truth, then I think that truth is correspondence with the Divine Intellect. But there are other theories of truth, besides the correspondence theory.
 
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