Aspies For Freedom

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!

I also have an interest in Mathematics, though my formal education in that subject seems to fall short of yours.

That having been said, I would suggest that the answer is not boolean (cardinal OR ordinal); rather, the nature of numbers is a dichotomy of cardinal and ordinal, with the question of primary reality depending upon perspective as opposed to being an absolute truth (akin to the concept of wave-particle duality from physics and chemistry).

Anyway, I hope that made sense.

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.

ATM: Yes, that would have been a better way to put it.  I concede.  Basically, I was asking about the *primary* nature of numbers, whether they are tied to physical reality or independent.  I opt for the latter.  

And Janet, I would never talk down to you, or to anyone else here.  I am reading some cool stuff.  I'll let it continue without interruptions.

B"H

To Simen:

Sir.  Thank you for reading my response.  I have so-far declined to answer your question of why I am a mathematical Platonist on-forum, to give a chance for the others to debate freely.  It seems that my posts are thread-stoppers at times.  After this response, I will give the others a chance to spin the thread out, so to speak.  

Let me explain to you, this once, why I reject (this one aspect) of Frege's, Russell's and Cantor's mathematical philosophy, while respecting them as thinkers who were well beyond me.  

It is really very simple logic to me.  I do not begin with the assumption that all mathematics is based on set theory, or on logical construction without any postulates that are self-evident.  If I *really* began with the former assumption, then I would assume a priori that there was a distinction difference between the "fiveness" of five elephants and the "fiveness" of five trees, since elephants and trees are different sets.  I would then have to "prove" that the same quality, five, applies to both.

I do not know enough about the set theorists to know if they would agree with my statement.  However, I am taking their initial assumptions, or lack of any assumptions, to a logically consistent conclusion (or premise).  In other words, whether they would have agreed with my assessment of beginning the argument on the basis of total skepticism that five stars are the same "five" as five donkeys, I believe that this is the only *consistent* basis upon which to build any kind of mathematical skepticism, not a set theory argument that really (implicitly) assumes the existence of numbers but then denies that one is doing so.  In other words, total mathematical skepticism implies that we have no common mathematical properties between any set that we define, even algebraic or logical properties that assume that one object plus one object in one set is numerically equivalent to one object plus another object in a separate set.

To me, this fails the "common sense" test.  I am willing to concede that common sense is more often common than sense.  But, in this case, I directly perceive a common "fiveness" to five elephants and five houses.  Instead of a physical object, "five" is perceived more as a dimension in my mind.  And, I believe that if you are honest, you will find that you perceive it the same way.  I would find it difficult to imagine that an intelligent person like yourself would perceive things in any other way, although I do recognize that many very intelligent people on this forum have dyscalculia.  Yet, the very term "dyscalculia" implies an understanding that there is a reality that one cannot perceive, just as sight-impairment would imply. And yes, the aforementioned room metaphor is a good one.  Light does not cease to exist because the door is closed.

In addition, I do not begin with the assumption that empirical reality is physical.  Your argument assumes that empirical data and physical reality are the same, excluding the possibility that one might directly "see" or "experience" numbers as what they are.  Now, you are free to be skeptical on this point.  However, numbers are part of my *empirical* experience, mind you, and I assume that they are part of yours.  Otherwise, as an intelligent person---and I make it a point to avoid calling my opponents "stupid" as seems to be the habit of some--- I do not see how you could have succeeded in mathematics without this understanding.  You clearly *HAVE* succeeded in understanding mathematics, so I have difficulty in seeing you as anything less than a full mathematical Platonist, in deed if not in ideology.  Like Frege and the other brilliant people cited above, you, an intelligent person with a clear understanding of mathematics, are not following through with your premises on a consistent basis.

I am a mathematical Platonist.  I cannot "prove" my argument using logic or set theory, because both of these assume numerical Platonism in an a priori sense, if one is really honest about it.  The argument that "one plus one does not equal two" when two clouds merge is one that is often made by mathematical skeptics.  However, this is really an example of two clouds becoming one, or 2--->1, not of 1+1=2.  The clouds would have to remain separate objects, or gas puffs, in order for 1+1=2 to be made manifest in that situation.  Otherwise, the conditions of 1+1=2 are denied.

So, while I cannot justify mathematical Platonism outside of the "self-evident" argument, which I freely admit, and which you are free to reject (although I think that most would not if they really thought it through), I believe that the other arguments are contradictory in that they cherry-pick their skepticism.  Most people would not accept one dollar in change if they gave 20 dollars for a fifty cent candy.  That would not be a mathematical skepticism.  That would be a confidence game.

I will also respond to you on some other statements that you have made on-forum, about other subjects, privately in PM.  I would rather discuss them with you privately.  On this subject you have made many interesting points, points upon which we disagree, but which still elicit my thanks.  I am glad that you have participated.  

All the best,

B"H

Simon, 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.  

As for the brain "creating," I thought that those who believed that the brain explains everything held to the view that the brain perceives, not "creates."  Is the brain the mind or not?

As for your third point, yes, I acknowledge that one *COULD* imply that capitalist exchanges are a game, one subject to social rules.  However, in this case, the rules are firmly based on mathematics.  I, for one, would not try to cheat a Vegas bookie out of his money on the basis of a theory that mathematics is a social construction.  I don't gamble, but you get me...Simple, and to the point.

OK, I hope that other mathematical Platonists will now take over.  Simen is smart, but I think that we have the truth on our side here so you guys can handle him while I handle the rest of my day!  Maybe I'll chime in down the road, but come on, you guys who perceive numbers can chime in, including synesthetics like myself.  

Thanks again, Simen and others who are participating.

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.

ATM: Simen, thank you for your response.  I can see that it will be a bit difficult to extricate myself from this thread!  I will pursue it a little longer, for as long as there is a learning process.  

Again, I never really made any assumptions about the brain and the mind being equivalent.  I was, however, noting a contradiction between the argument that the brain "creates" number sense and the notion popular among philosophical materialists that the brain "perceives" reality. Anti-Platonists have what seems like a contradiction on their hands.

If they argue that the brain perceives, then what of perceptions that do not fit their world-view, such as numbers, non-physical thoughts, et al?  On the other hand, if the brain "creates," then we have a problem with the notion of physical reality itself, being perceived by the brain.  How do we then know what is real, if it is perceived by an organ that creates?  If we go half-and-half, that some perceptions are true while others are false, then the burden is on the "half" folks to show what is perception and what is creation.  We can argue that one thing is reality and the other a dream, but we are not arguing with strict empiricism, but according to some form of Occam's Razor.  Occam's Razor is a philosophical doctrine that is problematic at times, since the "simplest explanation" is difficult to define, and not at all empirical.  My argument may seem like nonsense to you, but remember that to me a denial of mathematical Platonism contradicts my notion of what is self-evidently true.

In addition, you have used terms like "interacted with space-time."  This is a little bit complex.  The notion that numbers exist "beyond space-time" is very complicated.  However, I believe that numbers are real.  And, yes, I would say that in their fullest dimension they are beyond space-time.  I also believe that they are an important dimension of reality within space-time itself.  It would be best to describe the number-line as a dimension of reality, and leave any mention of the space-time continuum aside.  They may be beyond space-time, but it might be better stated that they are a dimension of reality in their own right, irrespective of our notions of space-time (Newton, Einstein, Lisa Randall, String Theory, et al).  

As for my invoking of "common sense," well, I think that the term is problematic.  It used to be that "common sense" settled an argument.  Today, that is not the case.  It is the closest term that describes my meaning.  However, "common sense" does not adequately fit what I am really trying to say, and the exact term I would use escapes me.  Perhaps the term "self-evident" comes closest, although I do believe that I should at least be able to show that I do not contradict myself, and that I have considered other world-views.  And, since I am learning that not everyone perceives numbers (dyscalculia is something I am only recently uncovering), I will not presume that you perceive numbers as I do.  

What I can say, however, is that your argument rests on a presumption that all empirical reality is material.  This has not been proven.  That is "common sense" to you.  It is not to me.  It appears, Simen, that both of our arguments rest on "common sense."  I will concede that I cannot prove my world-view.  However, I will argue that if one accepts logic, then one has accepted the number-line by implication.  Go over the rules of logic and you will see mathematics implied in them.  Even Frege and other theorists were really taking mathematical Platonism and simply re-arranging it.  I have seen the world-famous argument proving that 1+1=2, based on logic and set theory.  You can find it on-line (I'm sure that Simen has probably seen it, but some of you may not have.  Find it.  It's a good belly laugh!).  It often serves as a kind of joke.  Remember, however, that the argument really starts off with unstated assumptions that implicitly include 1+1=2 at their base.  In some sense these arguments become circular.  

The one really consistent argument against mathematical Platonism that I have found also denies empiricism.  Some argue for some kind of spiritual monism, or pantheism.  Friends of mine have been in these camps.  But, if you bring logic in to it, you are stuck with math.  2 is more than one is an important insight that, if denied, leaves us without the very structure of logic itself.

All the best.  I await your response, Simen.  I'm sure I'll get it. (-:

ATM: I see, you mean "the mind" and not "the brain."  I get it.  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.  

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.

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."

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.

Now, I really might go schizophrenic since this is getting the better of me!  All the best,

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.
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B"H

Simen, thank you again.  "Agnosticism" is honest.  I can accept that, even though I disagree.  I want to pause from posting for a while...maybe a day or two.  However, I *WILL* get back to you on the question of logic and mathematics.  It is interesting that you consider Russell's work a failure.  Did he?  

Personally, I read Principia and found it cumbersome.  The "proof," however, is a good laugh.  I am not sure that it "failed."  Honestly, I'm not arguing with you---I don't know enough about it. I am told that the effort spun off in to very practical applications, including radar.  I agree, however, that it failed to convince me of anything.

Again, let me get back to you on the logic and mathematics argument in a few days, because that is the only thread that I want to pursue.  Let me again state that I do not agree that the burden of proof is on me.  The burden would be on either one of our truth claims, that numbers exist, or that numbers definitely do not exist.  Agnosticism as a starting point is acceptable, although I think that it is untenable as a conclusion.  Since you have admitted agnosticism, then you have admitted that your premises are unproven (though believed by you).  All that is left is for me to show that they are inconsistent with the very construction of logic.  If I cannot, then I am left as much with a firmly held belief as you have, and we must part with a disagreement between gentlemen.

Until then,

ATM: OK, got it.  Let me take a little while to reply with more depth.  However, I would consider the use of the term "mind-independent" here.  I don't mean to be picayune, but could you expound on your meaning of "mind-independent" so that I can mull it over for next time?

It is no trivial question.  I know what the term means, but I want to know what it means to you.

All the best,

A fact or object is mind independent if its truth or existence does not depend on any mind. So, 1+1=2 is mind independently true, and an elephant has a mind independent existence, but taste is subjective, so it is not mind independent, and an oasis that only exists in a thirsty traveller's mind has no mind independent existence.
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ATM: I'm a little unclear.  Taste is subjective, so it is not mind independent.  OK, got it.  But, the taste itself is some type of chemical interaction involving the olfactory lobe (if I remember my brain map right), presumably "mind independent" in that it deals with the chemistry of neurons and not our ideas.  Thus, there is some boundary between the chemistry of taste-smell, as scientifically defined, and the interpretation of them that would define taste.  

Now, I guess that you have backed off (slightly) from equating brain and mind, at least for purposes of debate.  But, I can only assume that you *DO* hold the two as equivalent.  If I am right in that assumption, where would the boundary be between mind independent chemistry and mind dependent interpretation?  If it is all chemical, how do I actually know the difference?  If we deny brain-mind dualism of any kind, how do we then go back to it and call some things about our brain mind-dependent and other things mind-independent?  

When it comes to numbers, if they "work" in science, including the basic structure of our brain, then in what sense are they mind dependent?  Do they not then form some part of the construction of the Universe?  And, if they "work" in some independent sense, regardless of where we are in space-time, then are they not beyond space-time, even if in a very pragmatic sense?

And, by the way, I do not worship numbers.  I want to be clear that I am not a Pythagorean.  

Again, I'll get back to you on the logic=mathematics thread.  I can see that you have not made it a big part of your argument.  However, I want to consider Godel's Incompleteness Theorem and some possible objections before I put my foot in it.  I think I can justify my belief, but I might need to think about it.  And Guess Who, your post was cute.  I liked your ditty there.  (-:

All the best, and thanks to all who participated.

B"H

Thank you all for participating.  And please keep this thread going.  Lucie1, I would agree with you, although I think I might phrase it a bit differently.  Guess Who, that was a good one!  You also have a cute kitty cat, and that's a plus.  And Janet, yes, I strongly believe that numbers have a spiritual reality.  However, that is a personally held conviction that is beyond the scope of this thread.  Zakkie and others, thank you.  Your insights have been great.  And Simen, thank you again, and I appreciate you for participating.

Simen, my point in bringing up the whole "brain" question was the hope that we would get in to the hardware-software question a bit.  This was discussed by consciousness theorists a while ago.  Either the "brain=hardware", "mind=software" crowd has gone underground, or we just don't hear about it as much.  Perhaps I'm out of it.  But, I think that it is an interesting concept, one that probably leads nowhere here, but which is interesting to consider.

My deeper agenda, however, is to justify my belief that numbers are *BOTH* independent of mind, and beyond space-time.  The former is very difficult to "prove," because we do not have an accurate definition of what is beyond mind that we all seem to agree upon.  Frankly, it is a concept that is a bit hard to define.  We do not even seem to agree on what is beyond the brain.  

"Beyond the mind" is a meaningless concept because the four of us (however many there are here) cannot agree on it.  For both Theists and Theistic Pantheists, numbers would not exist outside of a Mind because there would be no "outside of the Mind."  All things would exist in the realm of being perceived, whether in the finite or the Infinite.  For adherents to other belief systems, by contrast, the brain is the mind, or something like that.  Therefore, in a sense, Simen, you and I do agree, albeit on different terms.  I too agree that numbers do not exist outside of Mind, but that Mind is capitalized.  I don't think that we will get beyond our respective points of view on this question unless we share common assumptions.

Now, for the question of numbers existing beyond material reality, including space-time.  That is really more of a focus that we can handle, since we seem to agree on what those things are.  Now, let us assume that Simen, Janet, Guess Who, Zakkie, and the others on this forum all share a similar understanding of space-time and material reality.  Let me also assume that we all perceive numbers.  I do not think that many of us on this thread have dyscalculia, and if you do, the very fact that you understand that fact is a tacit acknowledgment that there is a perception that you do not possess, a perception that is commonly possessed in society.  (As an aside; if I ever hit the jack-pot, causes related to fighting innumeracy will get my financial support)

Assuming that we all share a similar perception of space-time, most of us drive a certain number of miles to get to the store, pick out five loaves of bread, pay ten dollars for those loaves...In other words, we deal with numbers on a very practical basis.  If I were to say that one loaf of bread plus another loaf of bread gets us two loaves, we have a very practical way of saying 1+1=2.  That is sense perception, I agree.  However, it is a perception (or brain creation, if we grant Simen's claim, one that he has not demonstrated but which I will not argue at this point) that is not bound by space, time, or object.  One star plus another star gets us 2 stars.  One watermelon plus another watermelon gets us two watermelons.  Space does not change the rules of mathematics, nor does the type of object we use.  Frankly, even the curvature of space-time near a Black Hole would not change the fundamental rules of math that we would perceive.  One spaceship falling in plus another spaceship falling in gets us, well, two ships in big trouble.

In other words, in a very practical sense we can say that numbers are beyond space-time.  We do not have to get in to Quantum mysticism to get this one.  No time on a mountain top in Tibet is necessary.  Very practically, we can see that we operate as though math is beyond space-time, since it does not change depending on physical parameters.  The value of Pi does not change on the moon.  And, sadly, "e" does not change to one if a person is in debt because of a compound interest loan simply because they happen to live in another part of the country.  

Thus, numerical values are a dimension of *perceived* reality in their own right.  If we want to argue that we create those perceptions with mind, I cannot argue.  We might as well argue that we create space, time and objects in a similar way.  Actually, Simen, we can get in to a whole discussion of Theism, the Anthropic Principle, Quantum Mechanics and Consciousness some time.  For now, suffice to say that if we accept that we experience numbers, in whatever sense we experience them, we are directly experiencing a dimension of perceived reality that is not depending on space, time, or object.  

The very rules of logical construction assume homogeneity.  Logical arguments that begin with certain premises and end with certain conclusions would not change if we were to land on Mars.  In a sense, logic and mathematics are similarly beyond space-time.  "Socrates is a Greek, Plato is a Greek--->Socrates and Plato together are two Greeks" can be stated any way you want.  You can do it with set theory.  You can do it with logical arguments.  Or, you can posit the closure property of addition, and say that 1+1=2, with "Greeks" corresponding to the natural number-line.  I know that there are other mathematical systems in which 1+1=3.  However, if we are honest about it, what we have actually done is to tinker with the meaning of "+" or "=", changing our language, which does nothing to change underlying concepts.

Now, to the question of whether numbers go beyond perceived reality.  I don't simply mean beyond space-time, but beyond perceived reality itself.  In other words, full mathematical Platonism.  That is a complicated question.  Clearly, as stated, I have previously argued that they go beyond space, time and matter.  But, do they have an existence that is real in their own right, independent of those things?  Yes, I believe that they do.  However, as I stated to Janet, that goes beyond the scope of this post.  Hence, in answer to *why* I am a mathematical Platonist, I will be honest and say that I have my reasons.  They are for another thread, since they delve in to other realms.  What I think that *is evident from logic* is that numbers as perceived in physical reality are not merely creations of the mind, but fundamental to space, time, logic, and ultimately, via the "soft-ware" model, mind itself.

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.

I do not know how much I can keep posting an keep up with my class.  I might have to pause here.  If I write from now on, it might have to be shorter.

All the best,

ATM:  Yes, there does seem to be a problem here with definitions.  I think that what I will do, at some point, is create a thread in the "Time Out" section on mathematical Platonism, under "theology."  

For our purposes here, suffice to say that an admission that numbers are fundamental to the reality that we perceive is probably enough.  Simen takes existence to mean physical reality, which he automatically assumes is outside of Consciousness, while thought is "inside."  That may be common sense, but it is still unproven.  However, since he has admitted agnosticism, a position that I cannot argue against, I will now part from the thread.

Please continue, however.  I haven't read the article, so I'm gonna' hop to it!

No, it's not. That you refuse to admit it is another matter. If a number exists, then it would exist regardless of the existence of minds. 1+1=2 even if there are no humans. 1 exists, if it exists, regardless of whether humans think they exist. That much is logical. That much you yourself concedes.
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ATM: Ok, let me clarify my position, as I might have conceded to something I did not quite agree with.  1+1=2, as defined structurally in the simple definition of 1, +, =, and 2, without any "higher math" redefinitions, is true with or without humans.  Whether it is "independent of Mind" in another question; but unless you can *prove* that it depends on human minds, you cannot say that they are dependent on our existence since you have not proven it.

It seems as though the default agnostic position would incline toward their being real attributes of existence, unless proven to me creations of our mind, just as we generally assume that physical objects are real unless proven to be illusions.  If you want to assume neutral agnosticism on both counts, numbers and physical objects, by all means do so.  (Hume) I cannot argue that.  Nor can I argue against Berkeley---except to say that I never finished that Philosophy Degree I began, and I regret not being able to keep up with all of you pro's on terminology.  What I can say, however, is that, if we are honest, agnosticism is really a package deal with empirical reality and numbers, since numbers are empirical, directly derived from experience.

Now, I will concede to you the last word after this post, on this thread, and perhaps try to address a fuller discussion of Mathematical Platonism on the Time Out section, since I have been politely asked to post Torah/theology threads on that section.  For now, suffice to say, numbers and Nature go together so tightly (Golden Mean and Nature; Pi and circles; Fibonacci numbers and spirals, even the branches on a tree oddly enough) that I find it impossible to separate them.  One being a product of the human mind, to me, implies that the other would be.  I cannot prove this outright, as that is a bit hard to do in one sitting, but I suspect that as a scientifically educated person you would concede that science is impossible without math.  If one is imaginary, the other would seem to be also.  

Also, you mentioned "God's Equation" earlier.  It does not "prove" anything, but its use of e, i, Pi, 1 and 0 is kind'a cool!  Don't you think?

Let me give you the last word, Simen, since it seems like we're just simply not going to agree since we cannot seem to get to exact point-counter points or shared assumptions about existence.  And, please keep the thread going, Simen and the rest of you guys.  Also, please favor me by visiting my latest thread in the General section on favorite compulsions, leading to favorite slogans.  Basically, you post your favorite compulsion or interest, and then form a slogan around it.

Enjoy.  All the best.  Signing off thread.

B"H

Thank you all.  Perhaps we can conclude the debate, congratulate Simen for defending his point of view until the end, and leave this thread open for people who simply want to post the way that they see it, perhaps without a formal debate, just an open forum.  I will possibly address Platonism on some other thread, laying out some reasons for believing in it, as opposed to the realism that I have posited here.   The two positions are congruous today, but at one time in history they were VERY different.  I am a big "P" Platonist, whereas modern mathematical realists are small "p" platonists.  I might also address that link that Janet sent.  

Does anyone have a special interest leading to a possible forum slogan, perhaps not one that would appear on the webpage, since the Administration has not authorized it, but one that can be adopted by us for fun?  Please visit my post "Forum Favorite" in the general section.  Tell about your favorite obsession, and a slogan that could go with it.  

Thank you.

IMO numbers are abstract symbols that only "exist" inside the minds of beings that can think symbolically. You need a mind that can classify things into labeled groups (that is, take particulars and abstractify universals out of those particulars), this is because if you say "5 dogs" you need to be able to understand the concept behind universal "dog." Numbers are an abstraction based on quantifying things, and to quantify things a being must be able to use mental symbols to label groups of particulars.

Oh, and I'm a nominalist, I think universal labels only exist inside of minds and are constructs of those minds, and thus the numbers used to quantify particular instances of a universal are also constructs of minds.