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Does any language exist outside of the mind? What does it mean to "exist?" I believe that numbers are a created language that describes the universe as it is perceived by humans. It is the best language we have for this task, but let's not conflate description and actuality. I personally don't think humans have much of a clue regarding the universe's actuality.

Simen Wrote:
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.


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.

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.

If someone told me that they had a story of a dancing chicken in their mind, and they were not lying, the story would still exist - in the form of a series of words coded mentally.

If someone told me that they had a story of a dancing chicken in their mind, and they were lying, then the story would not exist. My belief in the existence of the story was based on a false premise.

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.

Simen Wrote:
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").


Point taken.

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.

Are these concepts physically manifest above the level of informational physicality? No.

So it all depends.

Simen Wrote:

EvilZakkie Wrote:
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.


This assumes that the nature of numbers is not an idea or a representation - I believe that this is exactly what a number is. The number 5 may be used a representation of, for example, 5 cows - but this doesn't mean that the number 5 becomes something other than the number 5 as a result. In the same way, the letter c may be used as a representation of the speed of light, but this doesn't stop it from also being the letter c.

In the example of the gunman above, the gunman doesn't exist, as the idea of the gunman is based on false data.

If you were to ask "Does the idea of the gunman exist?", the answer would have to be yes.

Simen Wrote:
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.


I don't see that as so bizarre - for example, if an airtight room exists, you could say that it "flickers in and out of existence" every time someone opens or closes a door.

In reality it doesn't - it just becomes something you'd have to use different words to describe.

With numbers, the concept of numbering exists, and it briefly becomes something you might describe as "the number 345678729435" when this number is used, then becomes something else.

Simen Wrote:

EvilZakkie Wrote:
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.


Point taken again - there's a few different meanings of the word.

I usually take the word as meaning "Thing to which great significance or symbolism is ascribed", but it can also mean such things as "non-physical existance", or even "form of religious 'energy'" in some cases.



Thought I'd throw a few more questions out there, just to clarify the definition of existence.

Does thought exist?
Does conflict exist?
Does happiness, sadness, or any other emotional state exist?
Do groups exist?
Does propoganda or rhetoric exist?
Does prejudice exist?

Simen Wrote:

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.


Yep, I got that - that's why I said "point taken" when you clarified what he ATM was talking about. To be honest, I'd jumped into the debate halfway through, and hadn't carefully studied the original post. I don't think either of us believe in abstract objects existing outside of space-time.

After this, you continued giving some "what if" arguments based on whether the word "exist" could be applied to numbers, so I continued the debate on that level.

I had a look through some info on mathematical platonism, and I thought it might be fun to argue the mathematical platonist point of view

(Note: my real opinion is that when talking of "objects" outside of spacetime, without physicality, it all just becomes word-games. Independant existence is such a broad concept that you could argue that all possible concepts have individual existence, or none of them - and you'd be right on both counts. Some word-games are enjoyable, though - and that's the spirit I'm entering into the debate from).

My thinking is that numbers do have an existence independant of mind, because once the concept has been described (i.e. once the concept has been "connected to"), the number has properties that are not controlled by the mind that contains it. For example, the number 8 has the property of being 2³ after the concepts of "2", "3", "8", and "power of" have been described - regardless of whether the person thinking of these concepts has ever thought "2³=8". If the person says "2³=9", for example, they can be shown that they are "wrong" - which shows that the number objects have properties not controlled by the mind creating them.

Of course it would be possible to assign properties to numbers such that 2³ does equal 9, but in this case you would be "connecting to" a different set of "objects", and everyone involved would quickly realise that a different set of objects was being talked about.

So there you have it - numbers have independant existence outside the minds of the people using them.

Simen Wrote:
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.


We seem to be getting back to our first debate. Are we debating whether numbers are mind independant, whether numbers exist, or the combo that "numbers have a mind-independent existence"?

The above arguments were to prove mind-independence, not to prove existence. To summarise the argument statements, numbers are mind-independent because they have mind-independent properties. You are correct that this cannot be used to show existence - I was not attempting to.

My earlier arguments were regarding existence (i.e. conclusion: depends on the definition).

I just went to the dictionary (http://www.dictionary.com) to see if there was an absolute definition, and it seems there's not:

Exists: 1. to have actual being; be: The world exists, whether you like it or not.  

Being: 1. the fact of existing; existence (as opposed to nonexistence).

Not very helpful. Without a working definition, we can't really debate existence at all - so before we get too far into that debate, what does the word "exist" mean to you?

I just wanted to reiterate this as a general rule for debates, as it's one that saves a lot of time, and is usually forgotten:

If you assert that x is y, and another person asserts that x is not y, or vice versa -  always go to the dictionary and look up x and y before continuing.

As often as not, the other person either agrees and is using an incorrect word, or is talking about an alternate usage of the word.

This case is a classic example - every single possible explanation of the word "exist" is possible, as the dictionary definition is circular. We will therefore either work out that we're using different definitions of the word, and then agree with each others statements, or we will find out that we are using the same definition of the word, and debate to a conclusion.

Simen Wrote:
There is a logical leap from "one spaceship plus one spaceship equals two spaceships" to "there exist numbers".


I'll re-iterate my earlier point - we still don't have a working definition of "exists", so we can't yet debate whether anything "exists".

Janet Wrote:

A True Monotheist Wrote:

If we argue that our perceptions are simply creations of the brain, then we can also argue that about material reality itself.  But, wait, the brain is a material organ perceived by itself.  What if that perceptions is illusion.  We have...a paradox.  And, Simen, that paradox must be addressed.  You have still not offered me an alternative to my notions based on brain chemistry...chemistry itself being defined by mathematical relations.


Optical illusions are but one example of this.  Our brains totally create an internal representation of the external world. sensory signals that reflect physical stimulus parameters are transformed into percept-based signals that reflect a person’s conscious perception. In most cases physical and perceptual aspects are highly correlated.  These sensory signals initiate a cascade of cellular and molecular processes in the brain that alter neuronal neurochemistry, and, brain structure and function.


What I found fascinating in the research is that we make up most of what we think we see! Just sort of fill in the blanks so that the whole picture makes sense to us. Really pretty extraordinary.

No, it follows that we have no reason to believe B.

And if B is analogous to, say, fairies, then we can say that we have as much reason to believe in fairies as we have to believe B, and since we positively believe there are no fairies, we should, to stay consistent, also positively believe that not B.
Or, to be precise, it would follow that C is no reason to believe B, and so, if there are no other reasons to believe B, then there is no reason to believe B.
I never said there was any logical contradictions in those ideas, only that we had reason to disbelive them. A coherent view is not always a rational view. The view that Mount Everest is seventy-five meters high is coherent (as opposed to the view that it is 2i-4 meters high), but it's not a rational view.

Do you not disbelieve the Flying Spaghetti Monster?
I repeat: I never claimed that it was a law of deductive reasoning, I claimed it was a principle of inductive reasoning.
Sigh. Dictionaries don't necessarily define domain-specific uses of a term. They're often incorrect when talking about the meaning of a term in a specific domain, and often, a term is thought up in some domain with a precise meaning, and then the term gradually becomes used outside the domain with a broader or different meaning that was once upon a time roughly analogous but isn't any more. In short, when talking about (say) mathematics, philosophy or scinece, we can't simply go to a dictionary and find a definition and then be sure that in mathematics, philosophy and science, this term is used exactly like it is used in general.

Let me quote Wikipedia, which has a more accurate description:

Quote:
Induction or inductive reasoning, sometimes called inductive logic, is the process of reasoning in which the premises of an argument are believed to support the conclusion but do not ensure it. Induction is a form of reasoning that makes generalizations based on individual instances. It is used to ascribe properties or relations to types based on tokens (i.e., on one or a small number of observations or experiences); or to formulate laws based on limited observations of recurring phenomenal patterns. Induction is employed, for example, in using specific propositions such as:

    This ice is cold.
    A billiard ball moves when struck with a cue.

...to infer general propositions such as:

    All ice is cold.
    All billiard balls move when struck with a cue.

In the same way that a scientific theory is never 100% certain. One can establish a very high likelihood of something's being true, so much so that it would be silly not to claim that we know it.

Therefore, you cannot claim that Occam's razor cannot give us any knowledge about whether or not there are numbers just because it's a principle of inductive reasoning.
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