Mathematics: Man-made or Divinely Created?

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What do you think – is mathematics a product of human invention alone or is there a divinely created substance behind it?
 
Divinely created. The Pythagorean Theorem is metaphysically true whether or not there is ever a physical right triangle.
 
Mathematics is a science–a body of knowledge which man knows, but he has not made the object which he studies, i.e. number and quantity.

However, that does not mean that number or mathematical principles are a “divinely created substance” This was Plato notion of real and substantial forms of knowledge which must exist in some non-corporeal “realm of the forms”. This is not the case. Mathemetics exists as real accidental forms that are part of other substances within creation. Aristotle makes the distinction (and St. Thomas echoes it) between substance and accident, and quantity is one of the categories of accident.

Thus, you can say it is a divinely created accident, but cannot say it is a divinely created substance.
 
Katholish said:
This is not the case. Mathemetics exists as real accidental forms that are part of other substances within creation.
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Interesting. But it seems to leave two possibilities:
  1. Numbers, as “part of other substances”, are properties of these substances. In other words, when you list the properties of a Maple tree, one of these properties would be that there are 457,357,434 of them (give or take a few). 🙂
  2. Numbers exist as accidental forms within the human mind. This has the classical problem that, if it is true, there are only as many numbers as any person has counted to. (So far as I can make out, being an accidental form has to do with something being instantiated, no?)
#1 is bizarre, and #2 is unsatisfying. But let’s build another option, a variant of #2:

2a) Numbers exist as accidental forms within God’s “mind”. This would make all of our arithmetic accurate (objective), and account for the fact that the solution to a problem no one has thought of is still true.

#2a works well, but I’m not sure how it maintains the “accidental” nature of mathematics. If numbers are within God’s mind, mustn’t they be substances? This answer is far more Platonic than Aristotelian.

(Please explain if I’m mistaken somewhere – I’m no expert in Aquinas or Aristotle).
 
you guys are making my brain hurt:D

I think mathematics is divenly created, look at Fibonacci sequences…
 
Prodigal_Son said:
Interesting. But it seems to leave two possibilities:
  1. Numbers, as “part of other substances”, are properties of these substances. In other words, when you list the properties of a Maple tree, one of these properties would be that there are 457,357,434 of them (give or take a few). 🙂
  2. Numbers exist as accidental forms within the human mind. This has the classical problem that, if it is true, there are only as many numbers as any person has counted to. (So far as I can make out, being an accidental form has to do with something being instantiated, no?)
#1 is bizarre, and #2 is unsatisfying. But let’s build another option, a variant of #2:

2a) Numbers exist as accidental forms within God’s “mind”. This would make all of our arithmetic accurate (objective), and account for the fact that the solution to a problem no one has thought of is still true.

#2a works well, but I’m not sure how it maintains the “accidental” nature of mathematics. If numbers are within God’s mind, mustn’t they be substances? This answer is far more Platonic than Aristotelian.

(Please explain if I’m mistaken somewhere – I’m no expert in Aquinas or Aristotle).
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You seem as best an expert as anywhere here.😃

God Bless
 
Prodigal,

To a degree, all three of the possible explanations you list are true and part of a complete answer. Since accidents cannot naturally exist independently, they must inhere in some substance.

Does the definition of Maple Tree thus involve number? No. In any given complex substance there are both proper attributes and non-proper (or accidental) attributes. Only proper attributes are used to define a things nature and definition. For instance, I am rational, tall, white, and living. Rational and living are proper attributes, whereas tall and white are not because they do not belong to human nature, but only to this particular instance of human nature (i.e. myself). Likewise, the number of trees is an actual attribute inhereing in a substance, but is not a proper attribute of that substance. I think that address the objection you have to the first possibility, though you might be able to come up with more based on my explantion.

As for numbers being accidental forms within the human mind, this is also true. That doesn’t mean that every possible number has to be explicitly considered though. Forms can exist in a latent manner within the possible intellect. Take for instance the form of triangle. You understand that a triangle is a geometric shape with 3 sides and 3 angles. That form is not in the human mind. Later on in the day, you recall that form and considering it further, you realize a proposition that is necessarily true about a triangle which you didn’t realize at first when you obtained the initial form. You now have realized that the internal angles must equal 180 degrees. You have learned something new, but do not possess a new form, nor does this imply that you didn’t really have the form of trinalge in the first place. It only means that you possessed the form without knowing everything there was to know about it, but the truth of the internal angles equaling 180 was an attribute of that form which was lying latent but was present because it was proper to the things nature.

All of the forms within the human mind are accidental forms, but since some of those are accidental forms of real substances, there are be accidental forms of other accidental forms within the mind. I realize as I write this that is sounds insanely complicated and I am trying not to confuse myself, but I am pretty sure that it is true.

As for the form being in the mind of God, everything that is is known by God, so I am not disputing it. My dreams are known by God, but that does not make them substances.
 
JMJ_coder said:
What do you think – is mathematics a product of human invention alone or is there a divinely created substance behind it?
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IMO, they are a product of human invention. But God certainly gave us the capacity to invent math, by giving us reason and placing us in a world that operates predictably.
.
 
JMJ_coder said:
What do you think – is mathematics a product of human invention alone or is there a divinely created substance behind it?
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Perhaps the human brain is an electrical muscle and as such uses binary code, the current flows linearly through the nerve cells and is either on or off; and so an argument is built up of sequences of binary type code. But perhaps the mind is different and attempts to use the brain to define and describe ideas and possibilities that are superior to and have allowed the possibility of a brain…:hypno:
 
Since I cannot seem to edit a post, I need to post a correction:
Take for instance the form of triangle. You understand that a triangle is a geometric shape with 3 sides and 3 angles. That form is not in the human mind. Later on in the day, you recall that form and considering it further, you realize a proposition that is necessarily true about a triangle which you didn’t realize at first when you obtained the initial form.
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That “not” should be “now”. It rather changes the meaning of the whole paragraph.
 
thing said:
…:hypno:
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…1101011 1110101001 10010101 1010101 10 101 01010101010 1010 100 1010010 001 10 10 010100010 10 1010 01 100110 0100101 0110 01 1010 010 1010 0101010010 0010 1010101 100100101 010 10100010101010001 010 100 00101010 100 10101 01010001
 
If there is such a thing as mathematical and logical truth, then there is such a thing as a being that is transcendent of physical reality. To speak of a mathematical truth, is to speak of something that was always true irrespective of our knowing or discovering it. Mathematical truth cannot be caused by that which changes or begins to exist or is limited in being, since an absolute truth is always true irrespective of that which is contingent. Also mathematical truth is not self existent, it is not a being, it is an abstraction of the order that we find in being, upon which other abstractions are made. Also, there is no truth in absolutely nothing. Therefore mathematical truth can only be a product of that which is eternal, as in a timeless transcendent unlimited being, since the truth does not change.
 
MindOverMatter said:
…1101011 1110101001 10010101 1010101 10 101 01010101010 1010 100 1010010 001 10 10 010100010 10 1010 01 100110 0100101 0110 01 1010 010 1010 0101010010 0010 1010101 100100101 010 10100010101010001 010 100 00101010 100 10101 01010001
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10 1010 01 100110 0100101 ?

:manvspc:
 
What an interesting conversation! As a Thomist-illiterate philosophy geek, I’m learning a lot. 😃
Katholish said:
Likewise, the number of trees is an actual attribute inhereing in a substance, but is not a proper attribute of that substance.
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But* in what substance *does it inhere? It cannot be an attribute of one particular tree – then, that tree would be both “one” and “many”. Is it an attribute of “treeness”, then, or the “idea/form of tree”, as Plato would have it? None of these make sense to me.
As for numbers being accidental forms within the human mind, this is also true. That doesn’t mean that every possible number has to be explicitly considered though. Forms can exist in a latent manner within the possible intellect. Take for instance the form of triangle. You understand that a triangle is a geometric shape with 3 sides and 3 angles. That form is not in the human mind. Later on in the day, you recall that form and considering it further, you realize a proposition that is necessarily true about a triangle which you didn’t realize at first when you obtained the initial form. You now have realized that the internal angles must equal 180 degrees. You have learned something new, but do not possess a new form, nor does this imply that you didn’t really have the form of trinalge in the first place.
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This latency idea is good, but it relies on the fact that the form of number is in the mind of God. After all, what is latent must be present somewhere – and, since mathematical truth precedes matter, the latency of math and geometry logically is made possible by its presence within the mind of God.

The whole conversation reminds me of Berkeley’s thoughts on the universe existing within the mind of God – which I do not, at present, endorse. But here’s an amusing tidbit:
Code: 
There was a young man who said "God
Must find it exceedingly odd
    To think that the tree
    Should continue to be
When there's no one about in the quad."
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Code: 
"Dear Sir: Your astonishment's odd;
I am always about in the quad.
    And that's why the tree
    Will continue to be
Since observed by, Yours faithfully, God."
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Substitute “number” for “tree”, and you’ve got our conversation. 😛
All of the forms within the human mind are accidental forms, but since some of those are accidental forms of real substances, there are be accidental forms of other accidental forms within the mind.
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Not sure I get it yet, but that’s OK. Is there an infinite regress here?
As for the form being in the mind of God, everything that is is known by God, so I am not disputing it. My dreams are known by God, but that does not make them substances.
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I’m not saying that God knows math. I’m saying that math is entailed by His nature, which is not the case for your dreams. We might say that mathematics is a property of God. (Gee, I hope that’s not heretical).😊
 
But in what substance does it inhere? It cannot be an attribute of one particular tree – then, that tree would be both “one” and “many”. Is it an attribute of “treeness”, then, or the “idea/form of tree”, as Plato would have it? None of these make sense to me.
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The attribute of quantity exists in creation as a whole in an indeterminate way. I was thinking about this point when I made my last response, but now realize I never mentioned it. It is part of the very nature of a material thing like “tree” that it exist in time and space. It is only in specific instances of a substance that there is a specific instance of some quantity. That being said, the potential for a whole range of quantities is a proper attribute of any given material thing (provided it comes in different sizes of course). A single existing tree is thus only one, but the form of tree includes the notion of potentially infinitely many.
This latency idea is good, but it relies on the fact that the form of number is in the mind of God. After all, what is latent must be present somewhere – and, since mathematical truth precedes matter, the latency of math and geometry logically is made possible by its presence within the mind of God.
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As I mentioned before, it is true enough that one a certain level everything depends upon the Divine intellect. Nothing in creation cannot exist without being held in existence by the Divine Intellect and Will. God is the first cause of all things, afterall–He is Being Itself. This primary causality is not exclusive, however, and that does not prevent the other two elements of our conversation (forms in matter and forms in the mind) from also being true.
Not sure I get it yet, but that’s OK. Is there an infinite regress here?
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No, because a substance cannot be an accident even though it can exist as an accidental form in the mind. There can be infinitely many forms in the mind, but there cannot be infinitely many “accidental accidental forms” in an “accidental substantial form”. [This choice of language on my part is probably not helpful, I apologize.]
I’m not saying that God knows math. I’m saying that math is entailed by His nature, which is not the case for your dreams. We might say that mathematics is a property of God. (Gee, I hope that’s not heretical).
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I have to entirely disagree. Mathematics is only true of the material creation which God made. Mathematics is entailed in the notion of material creation, but the creation of a material universe is most certainly not a necessary part of God’s nature–hence mathematics which is dependent on matter cannot be part of God’s nature.
 
Am I the only one fascinated with the concept of pi? It’s interesting that a circle’s diameter fits around its circumference exactly 3.16… times. However, the number of decimal places that pi goes to never ends (it’s irrational).

Anyone ever wonder why God didn’t set pi to exactly 3?
 
Reading through all the math talks is making me dizzy…

The way I see it is, math is a human expression to understand the divine design. Theologically speaking, it’s a way for man to know God by solving the jigsaw puzzle of God’s creation and to find that it is incredible, logical, and meaningful creation. Not like some say all existence is a random coincidence occurrence which in that case math would not exist because things would not add up for everything is just a random meaningless occurence.
 
Katholish said:
I have to entirely disagree. Mathematics is only true of the material creation which God made. Mathematics is entailed in the notion of material creation, but the creation of a material universe is most certainly not a necessary part of God’s nature–hence mathematics which is dependent on matter cannot be part of God’s nature.
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But surely you agree that the truth of mathematics or logic, is not caused by creation. Thus we must admit that it is true because of Gods “presence”, and is thus eternally true. For instance, imagine the sun as an absolute and the halo as the necessary and possible truths that surround the suns being in existence. It is because of Gods nature as being, and the intrinsic and perfect order within Gods being that allows the reality of mathematical truth from all eternity. In other words, mathematics is ultimately an abstraction of being. Do you disagree?
 
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