Aspies For Freedom - Nature of Numbers[Mathematical Philosophy]

A True Monotheist Wrote:

B"H

     Hello.  Thank you for reading my post.  As you can see from the "General" section, I have elected to stay on the list (for a time).  I would like to start a discussion thread, if you please.
     How do you view the primary reality of numbers?  In other words, are you a mathematical Platonist who sees numbers as being real in themselves?  Or, do you see natural numbers as being primarily ordinal, i.e. based on the counting of physical objects, as an infuriating friend of mine does.  This good friend of mine insists that numbers do not exist, that natural numbers are primarily ordinal and not cardinal.  We have had many heated, but friendly, discussions on the matter.
      I declare my belief that numbers are cardinal.  In other words, I believe that they have a reality in themselves that is not defined by set theory or some other mathematical abstraction.

What do you think?  My personal request for thread etiquette concerns these rules:

1) Please avoid calling someone "stupid", or use of bad language

2) Please try to cite appropriate sources for claims that are not easily verifiable, on-line or off-line

3)  Please avoid questions around whether you like mathematics or not.  I have placed "Mathematical Philosophy" in the subject heading.  I have done so in order to break my posts up in to "interest spheres".  My disinterested readers are free to move to other interests.  While I welcome you here, I do not feel insulted by your pursuit of those things that interest you, and would rather you pursue those things than attack those of us who do enjoy mathematics.

So, again, my question; "Are numbers 'real', or based on empirical reality merely?"  Happy discussions!

So you're asking whether mathematical realism is true, or whether anti-realism is true. Could we stick to asking whether numbers have mind-independent existence, or whether platonism is true, or something? Because when you write about numbers being "primarily ordinal" or "primarily cardinal", you're confusing matters. As you're no doubt aware, there are both ordinal and cardinal numbers. It gets confusing. Let's--to avoid any misunderstandings--just ask whether they exist outside of the human mind.

The answer to your question is no. I'm not a mathematical realist, insofar as I'm not convinced that there is reasonable evidence that numbers are anything but the inventions of our minds.

Janet Wrote:

A True Monotheist Wrote:

I declare my belief that numbers are cardinal. In other words, I believe that they have a reality in themselves that is not defined by set theory or some other mathematical abstraction.

I may be breaking one of you rules (about refering to liking numbers??), but I hope everyone adheres to the rule about not calling me stupid (which I am at risk), but.....(I feel so like a gate crasher here)

When one considers something such as the Golden Mean, numbers take on a spiritual quality. And I am saying this as an atheist. If I could ask you to speak down to me, does this magnificent organization of the world by numbers reflect a cardinal perspective demonstrating that they have their own reality??

No, it doesn't. The only "spiritual quality" numbers embody are those they evoke in your mind. The Golden Mean doesn't have any inherent "spiritualness", it's just what your mind chooses to attribute to it. Personally, I think it's neat that it shows up a lot of places, but I don't think of it as "spiritual".

Euler's identity, e^(i * pi) + 1 = 0, is also remarkable, but not "spiritual".

In any event, that the human mind attaches meaning to these numbers does not mean the numbers have any existence outside of our mind.

Since we've never observed any numbers, and won't ever be able to, I wonder how the hell we're supposed to know that they exist. Any takers?

Why do you believe numbers exist?

EvilZakkie Wrote:

Simen Wrote:

It's probably important to clarify what the word "exist" means in this context. For example, do the "inventions of our minds" exist? After all, they have a physical presence (comprised of chemical states and electrical neuron impulses).

For myself, my definition of existence is anything that where the evidence of the thing is not based on a false representation - regardless of it's physical presence. In this case, numbers exist, because numbers are a concept, and this concept does exist.

This is part of why I wanted to clarify. When I wrote "Could we stick to asking whether numbers have mind-independent existence, or whether platonism is true, or something?", I clearly meant that we should stick with asking whether or not numbers exist outside of the mind. That is what mind-independent means. And platonism contends that numbers are mind-independent (and also that they exist in some "other realm").

Further, when one reasons about numbers, one does not reason about the concept of a number. When we say that a system capable of representing arithmetic truths contains a statement of its own consistency if and only if it is inconsistent, we are not reasoning about concepts in the mind of a mathematician, we are reasoning about abstract objects. But do the abstract objects exist? I say not. They are but ideas in a mathematician's head. Very useful ideas, sure. Subjective ideas, of course not. mathematics is not subjective, yet numbers are not mind independent.

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For example, if someone told me a story about a dancing chicken, this story would exist, because the story is the words I heard or read - regardless of whether or not the words were in physical, verbal, or electronic form.

One representation of it would exist, but if we consider the story to be more than its mental representation, it does not exist. Numbers, as humans reason about them, are not conceived of as mental representations but as, well, numbers. And when you say that you have the idea of the number 2317 in your mind, that does not mean that you have the number 2317 in your mind.

When someone says, "It's only in your head", they don't mean that "it" exists. When a paranoid person thinks that a man is after him with a gun, but the man is imaginary, we do not say that the man with the gun exists, just because a representation of such a man exists in the paranoid person's head.

So we cannot say that numbers exist just because we as humans have representations of numbers in our heads.

Further, it leads to bizarre consequences like numbers popping in and out of existence as we think of them. If no one was thinking of the number 345678729435 while I typed this, that must mean the number popped into existence as I wrote it and flickers out of existence when I forget it until someone else comes along and imagines it.

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Simen Wrote:

The only "spiritual quality" numbers embody are those they evoke in your mind. The Golden Mean doesn't have any inherent "spiritualness", it's just what your mind chooses to attribute to it. Personally, I think it's neat that it shows up a lot of places, but I don't think of it as "spiritual".

All spiritual qualities are mental - spirituality is a mental concept. For example, if a god or gods existed, then they would be literal entities, and wouldn't have any inherent spirituality. Any spirituality directed towards these creatures would be entirely mentally based.

Religious people give spirituality metaphysical qualities. They believe that gods literally have some property called spiritualness, or some such.

EvilZakkie Wrote:

Simen Wrote:

Further, when one reasons about numbers, one does not reason about the concept of a number. When we say that a system capable of representing arithmetic truths contains a statement of its own consistency if and only if it is inconsistent, we are not reasoning about concepts in the mind of a mathematician, we are reasoning about abstract objects. But do the abstract objects exist? I say not. They are but ideas in a mathematician's head. Very useful ideas, sure. Subjective ideas, of course not. mathematics is not subjective, yet numbers are not mind independent.

Again, this comes down to the definition of existence. I can tentatively derive that from the above, your definition of existence involves "major" physical manifestation (major meaning, in this case, at a higher level than inforamtion-carrying physicality, such as chemical or electrical states).

Do the abstract objects exist outside of a particular persons mind?
Yes, the concepts have been written down many times, and are often coded into computer systems. It could be debatable whether the "understanding" of the concept creates the concept, but this term is debatable as well - for example, the computer "understands" the concept to the extent that it can use the concept meaningfully within limited parameters - most people are about the same.

The concept of the object is not the object. An abstract object is an object that is exists, but is not present in the physical world. They exist if and only if they exist outside of spacetime. The concept of the number one is not the number one, and so on. Mathematics cannot be reduced to reasoning about the brains of mathematicians.

I suggest you read some of the Stanford Encyclopedia of Philosophy's articles on mathematical philosophy, in particular this article. Hopefully that clarifies what kind of existence numbers are thought to have, etc.

A True Monotheist Wrote:

....

First: the "five-ness" is not something you directly perceive. The five-ness, if it exists, is nonphysical and acausal. Therefore, your physical brain cannot directly perceive it (indeed, a major problem for any platonist it to explain how abstract entities can be entirely nonphysical and entirely acausal, yet we know of them).

Second: the feeling of five-ness can easily be explained by appealing to your physical brain. Your brain creates the feeling of five-ness. That feeling does not exist outside the brain

Third: we can accept mathematics as useful regardless of the existence of the objects math is about. Math is provably useful, hence we use it. I do not have to accept any kind of change, since the principle of paying money already implicitly assumes the postulates of arithmetic. If nothing else (note the if), I am playing a game, and I will not have people break the rules on me.

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Simon [sic], thank you for your response. I can only say that you have made an assumption about what it is that I perceive. Perhaps I have made a similar assumption about you in my own post. However, I question your assumption that I do not perceive "five-ness" as a dimension existing in its own right. A good friend of mine agrees with you. However, you both have made an assumption about what I perceive that is problematic, and without proof based on any kind of empirical evidence as to what I perceive.

I don't doubt that you think that you perceive five-ness as a dimension existing in its own right, but that does not mean that you really do. Just like the paranoid will see men with guns who aren't there, so you may be seeing numbers that are not there. Since numbers are supposed to be nonphysical and acausal, and your brain is physical, there is simply no way you could perceive numbers that did exist. And since numbers are supposed to exist outside of spacetime and not interact with it, there is no way you could perceive them, since we're trapped inside spacetime. And even if numbers interacted with spacetime, there has never been any observation of this kind of interaction, so scientifically, we cannot do anything but assume they don't exist.

The only thing you've got is the appeal to common sense, but since you cannot find any logical argument that supports the common sense nor can you demonstrate any scientific evidence for it, we can only conclude that the "common sense" is not actually believed because there is anything sensible about it. I mean, there is no logical argument, nor any physical evidence, at least presented thus far, that is evidence for your intuition. Hence, I file it in the drawer together with the intuitive notion that the Earth is flat.

The properties of numbers are only true once we accept the existence of numbers. Things that do not exist have no properties. Hence, it would be circular to say that numbers exist because numbers have properties, without first establishing that numbers exist, so that they can have properties. Really, the has properties argument is never used for things we know to exist. There is no reason to say that elephants exist because elephants have properties that are mind independently true, because we have already established that elephants exist by observing them.

However, all you can say is that numbers, given their existence, have properties, and that is not very enlightening at all. Sure they do, if we accept the language of mathematics. I accept the language of mathematics, since it's very useful and I have an intuitive feeling that the language of mathematics is right. But that does not mean that mathematics is anything but a mental phenomenon.

Basically, I contend that numbers do not exist [outside of the mind], so numbers have no properties [outside of the mind]. The argument is valid. To deny it, one must deny the premise, that numbers do not exist. And one cannot simply do it by saying numbers have properties.

First, some general remarks about your previous post, Monotheist, that you wrote while I replied to another post, so I didn't get the chance to reply:

That the brain creates some perceptions is something you cannot deny, and that has nothing to do with platonism or non-platonism per se. We can all agree, I hope, that there are people who perceive there to be things or people that are not there. They may see aliens that don't exist or gunmen that are but illusions of the mind; they may imagine the shadows of trees to be people, or they may hallucinate the sight of an oasis in the desert where none is to be found.

If you think that the existence of brain illusions, that is, perceptions that are not the result of outside sense data but arise from the brain itself, implies that we cannot know what is true and what is not, then it is you that must concede solipsism. Hallucinations really happen. They are as much a problem for you as for me.

Moving on, it's simply false that logic has arithmetic embedded. I can use logic without assuming any of the postulates of arithmetic. In fact, one cannot do mathematics without logic, but one can easily do logic without arithmetic or numbers.

Further, it's untrue that arithmetic and number theory can be reduced to logic. This view is called logicism. Frege tried in the last part of the 19th century, Russell and Whitehead and Hilbert tried in the beginning of the 20th. They all failed. Logicism was shown to be untenable. The Principia Mathematica, from which the proof you talked about stems, was ultimately a failure, in the sense that it didn't achieve the goals it set out to reach.

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Well, your claim that the argument is "valid" is unproven. It is simply asserted, but unproven. And, I do not understand how one can prove a negative. If numbers do not exist outside of the mind, how would you ever know that? You sound very certain of something you claim to NOT perceive. The only way you could know the non-existence of numbers is to lay claim to a knowledge of the true properties of numbers, i.e. of their non-existence. That would then be a property that is known. But, again, even their non-existence is not really something that one can perceive outside of the mind, according to your argument, since there is no mathematical reality outside of the mind.

It's a straightforward application of logic:

That which does not exist, has no properties.
Numbers do not exist.
Therefore, numbers have no properties.

This is a valid argument.

The conclusion is true if the premises are true. The premise that numbers do not exist, of course, is controversial. But the burden of proof is on you to prove that there are numbers. I can explain why humans think numbers exist by appealing to evolution, without appealing to the existence of numbers. Hence, by Ockham's razor, I remove numbers from the equation. You need to prove the existence of numbers.

A True Monotheist Wrote:

[quote=Simen]
Basically, I contend that numbers do not exist [outside of the mind], so numbers have no properties [outside of the mind]. The argument is valid. To deny it, one must deny the premise, that numbers do not exist. And one cannot simply do it by saying numbers have properties.

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If numbers exist outside of the mind, and the mind perceives them, then we have a reality. If the mind merely "creates" mathematical reality, then we cannot know for certain whether numbers exist or not because, following your argument, we cannot know mathematical reality outside of our minds. That is a form of agnosticism that I can respect since it is humble. Yet, you deny *outright* that there is mathematical reality outside of the mind. Ultimately, the argument becomes solipsism, if really taken to its logical conclusion.

No, it's not.

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Since you are clearly not a pantheist or a monist, the only world-view that I could see you taking would be solipsism. That view would deny scientific empiricism as much as anything else. I will not attribute that label to you, mind you, since you have not claimed to be one. I am simply noting that it is the only argument a rationalist can take that would deny numbers and remain consistent within one's denial of realities outside of one's perceptions. Basically, one is following Hume, and denying any real knowledge. That is consistent, but inconsistent with any claim of ultimate knowledge, i.e. that "numbers do not exist."

I'm sympathetic towards physicalism. I am absolutely not sympathetic to any dualist views of the world. Why would you think I'm not a monist?

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My argument is self-evident to me. I can not claim that it will be self-evident to everyone. This is not the eighteenth century, and we cannot presume that we will establish truth by out-arguing the other guy. Now, I know the one limit to my argument, and that would be the "other" mathematical systems that have been devised. I will concede that point, and argue that mathematical reality is probably very multi-dimensional. I will not attempt to prove the existence of numbers, any more than I can prove this computer's existence in front of me. But I can say that the only rational argument about mathematical reality would be Platonism, agnosticism (we do not KNOW for sure, and never can), or some of the other philosophies that I have mentioned. In no sense can "there are no such things as numbers outside of the mind" ever be proven in a rational sense, or even held as self-evident to a rationalist since they deny the rules of rationality itself.

Are you agnostic about fairies?

See, I put the existence of numbers on the level of the existence of extradimensional fairies or gods. That is, I say with a very high confidence that they do not exist. However, if you insist that this is agnosticism, since there is a small logical possibility that they could exist, then I am an agnostic about the existence of numbers.

Let me clarify. My position is approximately this:

There is no satisfying reason to believe that numbers have a mind-independent existence.
All natural phenomena can be explained without appealing to the mind-independent existence of numbers. This does not mean that numbers are not indispensible to science, it only means that one can describe nature in numbers, but need not assume numbers are anything but a model we made up to explain some phenomena.
Therefore, we can with reasonably high probability, say that numbers have no mind-independent existence.

That, of course, is very simplistic, not meant to be a complete argument, just a summary.

Second to none has to be 1, because 2nd to 0 has to be 1 like 2nd to x is x+1. Almost axiomatic.

I feel like I am intruding, I have been reading through and a question that occurs to me it this .... could numbers be considered to exist as established empirical concepts? Surely then, numbers used as measures would be considered objective - rather than subjective.

Lucie1 Wrote:

Okay - I asked question - any thoughts on my question.

I will answer my question myself - numbers can be considered to exist as established empirical concepts - outside the realms of independent thought.

The fact that numbers are outside the realms of independent thought makes it objective - objectivity lends credence to the idea that numbers do in fact, exist.

Simen Wrote:

The burden of proof is on you to show that concepts exist outside of minds. I can't think of any experiment to show this. Can you?

The concept of numbers is expressed outside our minds by symbols representing quantity and measurements.

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