Where did I put those numbers?

I haven't read the entire thread and I'm throwing only two cents in here...

I believe, (and I recognize that beliefs can be kind of delusional), that ideas exist in time, without space, until discovered and/or articulated, and that numbers are a matter of semantics, while quantities are extant. Numbers are man made.

Certainly I would classify digits as man-made. However, one can talk of "oneness", "twoness", and so on.

...and therefore I say that, "oneness," exists independently of the mind, and so is discovered. The math is invented.

I think you may be confusing symbols being invented with Math being invented.

HAHA, if a tree falls in the forest does it make a sound!  Same argument.

People that are good at chess tend to be good with numbers.  Why?  Well it is the recognition of patterns.  Recognizing more and more complex patterns would make someone good at both fields.   Numbers represent a pattern.  The patterns are natural, the numbers are a basic way for people to understand the patterns and organize the patterns.  Languages have the concept lost in translation.   A phrase in one language does not translate into another language.  The question is almost the same for language. Do we think in a language or do we think and put our thoughts into a language.  The question would become if no languages were developed what would our thougths be like?  Numbers I think play this role in the mind to help organize the immense data that our minds can handle.   Data has to be organized, numbers allow us the ability to organize more data. 

@andrew1023: you should read "Number: The Language of Science" by Tobias Dantzig.. I have been peddling this book on another thread as well (I think) .. it is a popular science book, but it was acclaimed by Einstein as one of the finest popular science book he had ever read. =) ..

I do not doubt at all that symbols are the language of mathematics and science.

Well we're getting into increasingly muddy waters but from the dictionary I got

"Any abstract notion or idea by virtue of which we apply general terms to things."

for what a concept is.  An abstract entity would be a thing that exists in some way, the fact that they would be very strange should they exist works in my favour I think.  But of course there needs to be a story told to explain how we hold concepts if they are not based on abstract entities.  I don't have a good one to hand and as it would take some considereable study to formulate one I can't really take it any further.

Its an interesting area which seems to raise more questions the more you look into it.  We all think we have such a good grasp of what a concept or an idea or a number is but I find when I think about it I have little more than intuition backing me up.  This is not to tar you with my brush, Im only really speaking for me.  I'll have to get back into reading some philosophy me-thinks.

This reminds me of neurons being "neurological creations". Do you know of any other types of arguments for nominalism?

Did you mean to say it reminds you of numbers being neurological creations?  I was thinking along the same lines about ideas being certain patterns of electrical signals but then the question of whether these patterns can exist as particlar instantiations is raised and its back to abstract entities.  And Im sure there are empirical tests we could do to show that different patterns in two people corresponding to the same idea(of course various things could be argued here).  The philosophy of mind is an interesting but extremely limited subject.  Consciousness is a strange phenomenon indeed. None of this makes me want to turn to dualism or religion though. 

The only other arguments 'for' Nomanilsm that I have encountered are arguments against Platonism.  It was always the problems with abstact entities that swayed me to nominalism rather than a strong argument for the nominalist position.  Then again if there's only two options an argument against one is an argument for the other, possibly.

To St. Augustine, such numbers were products "of the Mind of God", just as an aside note.

Lucas-Penrose Thesis.

I have scanned this thread and would like to make a few comments to argue my philosophical point of view. I could be categorised as a neoplatonist. I believe that the entire structure of mathematics is true independent of our physical universe. For example, I believe the notion of the integers as an abstract structure for dealing with counting discrete objects would have the same abstract truth, the same theorems, etc. regardless of what the laws of physics were, and the same goes for the entire edifice of pure mathematics (much of which finds application in the physical world).

But I would also point out that pure mathematical facts can be seen exhibited in real world instances. For example, someone said while 7 was prime, 7 apples were not prime. Not so, I claim. Here is a statement about apples:

"If you have 7 apples and put them into a number of separate heaps, each of which has the same number of apples then either there is only one heap or all the heaps have only one apple."

The same statement could be made about any prime number of apples, but not about a composite number of apples. Seems to me the primality of 7 is exhibited very concretely in a real world instance.

We have a neoplatonist in the room! If you please, how do you differ from us more "traditional" platonists? I don't know how many of your positions or if there are sub-categories of your kind, etc.

Basically the same except we are newer.  When I wrote neoplatonist I intended it to mean a school of thought that included key ideas of Plato but not all of his beliefs. Unfortunately I see someone waylaid the term in the 3rd century AD, so I may have to call myself a neoneoplatonist or a pseudoplatonist to avoid any unintended connotations.

Anyhow, one aspect of my point is that if intelligent life arose in any part of this universe or another with different physical laws, they would develop concepts of counting and would at some time get to the theorems of number theory, because cardinality and all the facts about it are absolutely universal (in a broader sense than merely applying to our universe). The patterns of mathematics that we discover and find ways to represent are "natural" in the most fundamental way possible (I know of nothing else so universal or so fundamental).

While it is possible to imagine a universe where all physical laws are different, it is not possible to imagine one where the laws of cardinality are different.

When mathematicians call themselves "Platonists" they generally don't mean that they subscribe to all of Plato's beliefs, or even most of them.  What unintended connotations do you mean?  

That usage would happily cover me then. I believe that the patterns we have discovered in mathematics are a form of absolute truth that exists without any reference to the physical universe, and indeed are probably the reason the universe exists. I would put it differently to Plato and never liked the references to some ghostly world of universals (probably just a matter of semantics and translation). As a specific thing I don't subscribe to, I would give the example of Plato's doctrine of recollection, which heads unnecessarily in a speculative direction similar to what religions do. But then, I am not sure that Plato would agree that the patterns of mathematics are the reason the universe exists either. :-)

I think Andrew 1023 is right numbers serve us just to make our life easier, let's say if you would "draw" fibbonachi numbers then you would get a spiral, and that is understood everywhere

I do not know if anyone is still tracking this topic, but I looked back here to refresh myself.

It appears to me now that extreme intuitionism, at least time dependent intuitionism, (Brouwer's version) is circular. In the meantime I have also run accross another realist account of mathematics called Structuralism (see Shapiro), which has been argued against using set theory...from what I have read, structuralism is more natural for something like group theory, but may be incompatible with set theory. I do not know much about Dummet's anti-realism. I think Imre Lakatos is the most interesting so far, but partially misguided.

I also think that most "problems" with platonism are really just questions about the consequences of platonism. And I think existence proofs can be supplied to show that such questions can be answered. For example, on materialism, platonism has no problems!

(1) Assume materialism is true

(2) Then brain states reveal mathematical insights

(3) Assume that the brain can be sufficiently mathematically modeled

(4) Therefore there is a mathematical account of how we can know causally inert things

Or, one can adapt a dualist response to an objection against dualism: We know that intents can cause changes in another person's behaviour, such as since I think my friend intends to do something, I will base my behaviour from that. This intent cannot be physical, because if that were true he would be physically compelling me in some way to act. But we can be entirely physically disconnected. What this means is that we have a different type of causation that can analogously lead  us to recognize mathematical statements in the same way.