Are Numbers Necessary?

Abecedarian

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

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

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

↪BrianW

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

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

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

↪Abecedarian

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

↪Abecedarian

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.

Queen Cleopatra

↪Mentalusion

:up: Quite the exposition.

Kippo

However, possible worlds are conceived as discrete entities — Mentalusion

And who conceives them - us!

MindForged

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

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.

Mentalusion

↪MindForged

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. — 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.

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. — 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.

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. — Mentalusion

Something about this seems mistaken. You say an incomprehensible world be would chaos, but is that a preclusion to existence?

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. — 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.

TheMadFool

↪Abecedarian

There are uncountable things in our world. How does one count love or hate?

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.

unenlightened

Here's a little article that might be of interest.

https://www.prospectmagazine.co.uk/magazine/kurt-godel-and-the-romance-of-logic?fbclid=IwAR0m6ifJB0TLm2IdreqB3Wt8Ig6iRB9a9LjU3IgA4PlhnV28kLgNHuIF8nw

Pattern-chaser

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. — BrianW

OK, so which part have you recognised as being false?

Walter Pound

If numbers didn't exist at all, it is hard to see where any logical contradiction would arise.
Mathematical fictionalism would be dead in the water if they were.

Pattern-chaser

↪Walter Pound

If numbers didn't exist at all, mathematics would be dead in the water, wouldn't it?

BrianW

OK, so which part have you recognised as being false? — Pattern-chaser

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.

Relativist

↪Abecedarian

What is a concept? I suggest it is an abstraction, described propositionally.

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.

Pattern-chaser

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. — BrianW

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.]

BrianW

Where in the scientific space-time universe is "two"? — Pattern-chaser

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.

Walter Pound

↪Pattern-chaser

No, mathematics would still be something that mathematicians could engage in. Numbers don't have to exist for a mathematician to say that the square root of 2 is irrational or that 2 is equal to 4 divided by 2.

Banno

Seems to me that this thread is based on a fundamental misapprehension about possible worlds.

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.

Pattern-chaser

Numbers don't have to exist for a mathematician to say that the square root of 2 is irrational or that 2 is equal to 4 divided by 2. — Walter Pound

:chin: :wink:

Walter Pound

↪Pattern-chaser

2 = 2 is true without 2 being a real entity.

Pattern-chaser

2 = 2 is true without 2 being a real entity. — Walter Pound

You didn't say "not real" you said not-existing:

Numbers don't have to exist for a mathematician to say... — Walter Pound

With no numbers, there can be no number theory. Your position is unjustifiable. Sorry.

Walter Pound

↪Pattern-chaser

you are confusing two important things.

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.

Pattern-chaser

The symbol "2" is not itself 2... — Walter Pound

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.

Walter Pound

↪Pattern-chaser

if there are no numbers that exist in reality, then mathematical fictionalism is trivially true.

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.

Are Numbers Necessary?

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