Are Numbers Necessary?
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.
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.
Comments (31)
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.
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.
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?
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.
:up: Quite the exposition.
And who conceives them - us!
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.
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.
Quoting MindForged
I mean as little as possible by it so as to leave open to interpretation what it could mean. Most generally, I take just to mean "capable of being understood" whether by humans or any other rational creature. My intuition for its truth is that any "world" that were completely incapable of being conceptualized in any way by any kind of rational creature would not qualify as a world at all. It would essentially be chaos.
Quoting MindForged
Agree about the math formalism, but I don't see why the premise wouldn't work with whatever theory of numbers you accept, whether platonic, intuitionist, whatever. Are you suggesting it makes sense to suppose there could be different possible worlds where, say, constructivist , platonic, etc. interpretations hold in each? I guess my assumption is you would first have to decide what you think numbers are before running off to look for them in different possible worlds.
Something about this seems mistaken. You say an incomprehensible world be would chaos, but is that a preclusion to existence?
Quoting Mentalusion
Sure, why not? It's fairly common (for logicians anyway) to speak of worlds where different logics obtain. If this couldn't be done I don't even think such logics could be given semantics. Now this is the sort of thing I meant when I said the world's might well be incomprehensible (in the sense of running very counter to the intuitions our world has shaped) and yet still have some kind of existence.
However, we're also very quick to compare things. Who loves me more? What hurts more, the punch or the kick?
In such instances numbers become essential. It brings accuracy to our discourse. On a scale of 1 to 10 my pain is a 10 and yours is a 1.
https://www.prospectmagazine.co.uk/magazine/kurt-godel-and-the-romance-of-logic?fbclid=IwAR0m6ifJB0TLm2IdreqB3Wt8Ig6iRB9a9LjU3IgA4PlhnV28kLgNHuIF8nw
OK, so which part have you recognised as being false?
Mathematical fictionalism would be dead in the water if they were.
I no longer think numbers were invented. I'm now inclined to think numbers are an expression of a relationship which has always existed in nature, and which we discovered. From my point of view, I think the significance of mathematics is something inherent in nature or the workings of reality. It's like, from the moment we chose to use numerical language (which seems to be a fundamental mode in reality) we had no choice but to arrive at mathematics in one form or another. That said, I find it to be very complex to use, especially in philosophical expressions.
You seem to be suggesting there exist platonic entities that correspond to numbers, but that our propositional description of these things is flawed as a consequence of our intellectual limitations. Without a referrent, there can be no flaw in that description.
Where in the scientific space-time universe is "two"? Yes, you can find similar (is there any such thing as exactly identical?) things there, which we might enumerate, but where is "two"? It isn't there. There is no logical reason to subdivide the universe anyway; it's one thing. Thus we need only the number "one", and even that is a concept we invented, as it doesn't exist in the universe.
Yes, we can say that these things (help us to) express things about the universe, but that doesn't mean they exist in the universe. Perhaps it is more helpful if we refer to all of these things ( the things like numbers that we invented) as 'maps' that we have made to help us navigate the universe? [No, "navigate" not meant literally.]
Two (or any other number) is a very specific and distinct condition and relation in the universe (or in reality). We could call it by any other name but the exact significance of that identity can never alter. I think numerical values exist because, otherwise, it would be impossible to account for relativity or the many aspects of reality.
They are stipulated, not found.
Such stipulations are in terms of individuals and their properties.
Can you stipulate a possible world without numbers? No, because numbers are neither individuals nor properties of individuals.
So what we have in this thread is another fine example of how language misleads.
:chin: :wink:
You didn't say "not real" you said not-existing:
Quoting Walter Pound
With no numbers, there can be no number theory. Your position is unjustifiable. Sorry.
Mathematical platonists will argue that numbers are real entities.
Mathematical fictionalists argue that they are not real.
You can say that 2 is identical with 2 and still argue that 2 does not exist. The symbol "2" is not itself 2 so there is no contradiction in mathematical fictionalism being true.
No, it's not. But it represents 2, the number. If numbers don't exist, 2 does not exist, and the symbol ("2") has no meaning, for there is nothing for it to symbolise. Your position is untenable and ridiculous. Please drop it. Thanks.
What you are trying to say is that if numbers doesn't exist, then symbols, such as 2 and 4, don't correspond to anything so therefore mathematics is impossible, but it isn't.
The mathematician who argues that the square root of 2 is irrational can simply argue that this is true by virtue of the definition of 2. The mathematician is not obligated to say that the square root of 2 is irrational because of some fact of the universe or of the real world.
Impossible to account for? Impossible to/for who? Humans, of course. The universe has no need for numbers, which is lucky because they don't exist in the space-time universe.
But numbers sure as Hell do exist, just as Harry Potter does. In the minds of humans. We created them. Harry we made to entertain us, and he does so admirably. Numbers we made to help us describe the space-time universe. Because we can't grok the universe in one piece, as it really is, we split it up into smaller and smaller parts until we stand a chance of the merest whiff of understanding. It's what we do, because we have no choice. But what we do is out of necessity, and it has no logical or rational justification other than that: we have no choice. Numbers help us describe the universe, but they are not part of the universe.
As for these 'specific and distinct condition and relations', these too are human inventions. The universe is what it is, and it is able to do what it does with none of this human baggage you are dragging into the mix. Some parts of the universe were attracted to one another billions of years before Newton conceived of "gravity". The attraction the universe just does, all on its own, because it cannot do otherwise. But it doesn't need 'gravity' to do it. It's us who need 'gravity', because without it we cannot clarify or describe our understandings to ourselves or to one another.