• Abecedarian
    13
    I wanted to introduce a new idea into the field regarding humans limited knowledge, God’s omnipotence, and the existence of simple concepts such as numbers. The argument that I am presenting is in response to those who believe that numbers necessarily exist. I believe that it could be the case that are limited understanding as human beings does not allow us to fully understand certain concepts. Humans already do not understand certain ideas about numbers such as the idea of infinity and its various paradoxes. Although people cannot fathom these ideas, I have no doubt that God can understand infinity and its implications due to His omnipotence. I would take it even further to say that it is possible that God’s understanding of infinity is different enough from ours that our concept of infinity has little truth to it. I have formulated my argument as follows:

    1. It is possible that beings of limited cognitive ability (like humans), do not fully comprehend even the simplest concepts in comparison to beings of infinite knowledge such as God
    2. If premise 1, then it is possible that those beings of limited cognitive ability could be in error of the simplest concept’s nature and even of the concept’s existence
    3. Therefore, it is possible that beings of limited cognitive ability could be in error of the simplest concept’s nature and even of the concept’s existence(1 & 2 MP)
    4. Numbers are one such concept
    5. Therefore, it is possible that humans are in error of number’s nature and even of their existence.

    I think that it could be the case that the knowledge gap between humans and God is so great that our explanations or perceptions of certain ideas could be far from God’s complete understanding.

    One objection that I have heard is that numbers are logically necessary due to it seemingly being impossible to imagine them not to be. Opposers have said, “Try to think of God. Even the perception of God demands the concept of numbers as you will have to ask yourself, ‘how many gods are there?’ or ‘list god’s characteristics’”. It seems intuitively true that objects, ideas, and even concepts cannot exist without numbers. However, in my argument, that would make perfect sense due to our low level of knowledge. Of course, we would be unable to come up with a counterexample of a non-existence of numbers as humans are vastly limited in our cognitive abilities.
    Through this argument, I am not saying that numbers do not exist, but am merely illustrating the idea that it is possible that numbers do not necessarily exist.
  • BrianW
    481
    Since joining this forum, my ineptitude to understanding mathematics has been overwhelmingly clear. Before, I would have said that 'mathematics is the science of numbers'. I used to think that numbers were invented components of the numerical language which mathematics used to express logic or principles of nature. Now, I think part of it is true. But, I still can't figure out what abstract mathematics is all about.

    As to the level of cognition by beings, I think logic dictates that the little we know may still be correct even if it's partial. Perhaps, it like a driver who knows how to move a car but doesn't understand the inner mechanics of what makes the car move. I think, while partial knowledge is necessarily incomplete, it is not always wrong/in error.
  • BrianW
    481
    Through this argument, I am not saying that numbers do not exist, but am merely illustrating the idea that it is possible that numbers do not necessarily exist.Abecedarian

    Sometimes I want to share this sentiment but, when I look at how symmetrical nature is or has been even long before humans came into being, I'm not so sure anymore. From what I can tell, numerical relations are like logic or laws of nature, they've existed since the beginning.
  • Abecedarian
    13
    I also cannot understand how numbers or logic would not be able to exist in this world. But just because my own understanding cannot comprehend it does not necessarily make it true. In fact, I feel like it would be natural for humans to not understand what could be far beyond their abilities.
  • Fuzzball Baggins
    12
    I suppose it is possible that our concept of numbers is wrong, but saying that humans are not omnipotent and therefore don't understand everything doesn't make it likely that we are wrong about numbers, in my opinion. Just possible. It's important to keep the mind open to all possibilities. But we have no evidence that our understanding of numbers is wrong, so by the law of occam's razor I think it's fair to assume that numbers are real until some evidence shows us otherwise. I also think that if we were to reject the existence of numbers we would need some alternative hypothesis for why putting an orange next to another orange results in a group of oranges.
  • Abecedarian
    13
    It was not my intent to try to disprove numbers. I think that numbers exist as well. I simply am counteracting the argument that numbers NECESSARILY exist.
  • Mentalusion
    64


    1. Any possible world that is intelligible is such that it contains some structure and form.
    1a. All possible worlds are intelligible
    2. At least some aspect of all structure and form is inherently quantifiable
    3. Anything quantifiable is capable of being expressed numerically
    4. All possible worlds contain aspects that are capable of numeric expression
    5. The capacity for expressing numbers is sufficient for numbers to exist (whether or not anyone uses them or has discovered how to use them)
    6. Numbers exist in all possible worlds
    7. Therefore, numbers necessarily exist

    If you agree the above argument is valid, which of the premises do you think is questionable?
  • Mentalusion
    64


    Here's another:

    1. Necessity is determined by truth in all possible worlds
    2. However, possible worlds are conceived as discrete entities
    3. Discrete entities are countable
    4. Counting requires numbers
    5. Therefore, the concept of necessity (necessarily!) implies (because it assumes) that numbers exist.
  • Kippo
    46
    However, possible worlds are conceived as discrete entitiesMentalusion

    And who conceives them - us!
  • MindForged
    546
    1. Any possible world that is intelligible is such that it contains some structure and form.
    1a. All possible worlds are intelligible
    2. At least some aspect of all structure and form is inherently quantifiable
    3. Anything quantifiable is capable of being expressed numerically
    4. All possible worlds contain aspects that are capable of numeric expression
    5. The capacity for expressing numbers is sufficient for numbers to exist (whether or not anyone uses them or has discovered how to use them)
    6. Numbers exist in all possible worlds
    7. Therefore, numbers necessarily exist
    Mentalusion

    1a is probably false, depending on what you mean by intelligible. Lots of worlds are presumably unintelligible if their structure is such that it runs very counter to our own universe. So say a universe where objects are clearly distinguished would probably appear as very unintelligible to most or all people. But if by "intelligible" you mean "coherent" (i.e. not logically trivial) then 1a is true.

    2 & 3 are suspicious because there are different ways of assigning quantity to thing. Numbers are not the same thing across all mathematical formalisms, and so it does not follow that some numerical system that is apt to one particular possible world is applicable to all of them. Constructive mathematics and classical mathematics - not to mention Paraconsistent mathematics - look quite different. Some numbering systems lack entire types of numbers. Standard, classical mathematics doesn't have the hyperreals, for instance.

    Anyway, talking about the necessary existence of numbers (this is aimed the OP and Mentalusion) in the same way one does for non-abstract objects just sounds wrong. "Existence" in mathematics is very different than the colloquial and philosophical use of that term. This seems relevant since lots of different abstract objects that correspond to our notion of numbers and such might come out differently depending on the math you're using.
  • FordFestivaPhilosophy
    8
    Your defense for premise 1 ( It is possible that beings of limited cognitive ability (like humans), do not fully comprehend even the simplest concepts in comparison to beings of infinite knowledge such as God) seems to be rooted in some lack of the extent of our knowledge. Doesn’t it seem, that there must be some certain simples necessary for cognitive ability. You say that you accept that numbers do, in fact, exist, but how do you know this? If, as you say we are unable to know even the simplest concepts, then how are we supposed to know anything more complex? Your argument is more of a reason to doubt all knowledge, as, if we cannot even understand these cognitive simples, such as numbers, and logical truths, how then can we know anything at all? It would seem, your argument would seek to unfound any knowledge claim.

    Also, there seems to be an implicit premise, which I am not sure one should accept: if there are limits in one’s cognitive abilities, one must accept that it is possible that all knowledge is impossible. Well sure, I suppose it is possible, but I would question the probability of such a sentiment. For instance, I would claim that numbers exist. It seems that there are these concepts which so accurately predict phenomena in the real world, that their existence is highly improbable. Fictions would almost certainly not be a certain and predictable as mathematics would indicate numbers are. However, despite the probability that numbers do in fact exist, your same argument would argue that it is possible that it is entirely fictional, and that we understand nothing. Because we have limited cognitive ability, is possible that we are wrong about absolutely everything, even, as you say, the simplest concepts. Should we begin to doubt the simplest concepts then? Surely not, for then how would we ever think we could construct meaningful thoughts and beliefs?

    Thus, I believe, your premise one (1) is a much broader claim that at first it seems, and when taken to its logical extremes, results in some absurd, or at least improbably claims, that I’m not sure you meant to endorse.
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