NUMB3R THEORY

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NUMB3R THEORY

Post  JWSanborn on Thu Nov 01, 2012 11:22 pm

A few weeks ago, during our weekly Philosophy Club meeting John (Chris-- In fact, since his name is not well-defined, I will henceforth refer to him as JCB, whether he likes it or not.) led a brief discussion about the existence of abstract concepts such as numbers (at least I think this is what it was about I zoned out) which got me thinking. My recollection is that JCB holds the position that Abstract concepts do not exist and that this is no problem for scientists or anyone whose work relies on abstract concepts. To be glib, numbers aren't real but your bridges won't collapse. I am only now beginning to formulate my opinion on this matter, but as it unravels I shall share it with you.

To begin with I sincerely believe that JCB's position is the most coherent of the positions he stated as options, but I sincerely believe the positions he stated were only a (proper) subset of all the reasonable positions, and he left out the most obvious and coherent of all. That position is that numbers do not exist as abstract concepts, but they do exist-- Numbers are concrete objects in the real world.

Believe me friends, I understand this revelation can be quite baffling at first, it certainly was for me. What does it mean for numbers to be concrete? I will tell you. Every number (with the exception of the non-real complex numbers) has some physical representation in the physical world. It is a fact that is overlooked by most people, as when you are not looking for it, it seems impossible that a street lamp is actually 1.5783 or that a brick might be the number 9. But once you've accepted this position it is quite reasonable.

You may still be skeptical, and I accept that. As a skeptic myself I understand your need for proof. I've dedicated my life to the search for the “real” numbers in order to prove this position. I am going back to the very foundations of mathematics to give me clues to the location and form of these numbers. Of course some numbers I expect will be natural, represented through only the nature of the universe. Some I expect to be created at the hand of man. Others still, specifically the non-real complex numbers I believe I will only be able to discover through deep meditation and imagination. This will be an arduous journey, and I realize that no matter how many numbers I may find, it will never be enough, but I hope others will carry on in my work, and someday, we will have all of the numbers.

Progress update: I have discovered that Happy primes come from happy numbers, and happy numbers come from California. (it's a link!)

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Erewhon

Post  John on Sun Nov 04, 2012 10:51 am

Obviously, I am pleased that my discussion has led to a rival hypothesis which is so much more tenable than the current fare (abstract numbers? really? what does that mean, are they like a summary before the rest of the numbers?). Mr. Sanborn, to whom I will refer as JaSanborn only if he dislikes it, has pointed out a complexity which, with some circumspection, I believe I can answer briefly. The answer, of course, like most facts, is a posteriori, since there is certainly nothing inconsistent in his a priori assertion that the numbers formerly known as abstract (now known simply by the pretentious symbol "ℝ" and also refusing to let their cover of Coldplay be played on Youtube), are indeed concrete. My proof trades on his search update, and so is particular as opposed to general, but why not generalize anyway:

(1) All happy numbers are happy primes
(2) All happy primes are California residents
Conc: All happy numbers are California residents

Now, one representative of the happy numbers is 7, and from the argument above, 7 must be a resident of California. It is a truism, however, that residency cannot be held between states. 7, unfortunately, seems to have residence in Connecticut, Oklahoma, and Oregon under the heading of 'electoral votes.' Note that California is not on the list, and that 7 is committing electoral voting fraud. Even worse, this seems to be a common occurrence in the electoral college, since several other numbers seem to be represented in multiple states. A Congressional Report will certainly turn up overlap, in what the 24 hour news cycle will inevitably refer to as Numbergate. What to do?

The options seem clear: Either we must conclude that happy numbers do not reside in California, thereby refuting JaSanborn, or we must abolish the electoral college, thereby refuting JaSanborn (since the sixth paragraph of his posting defends this position as well). Alternatively, we could watch this video and grab a juice box...
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Re: NUMB3R THEORY

Post  JWSanborn on Sun Nov 04, 2012 2:12 pm

My distinguished colleague JCB must be pretty pleased with himself right now, considering he probably believes he has "refuted" my claim. Allow me to remove that warm feeling you've got there, JCB; You're wrong.

First of all, your argument in no way reflects what I was saying, a correct argument would read as follows.
(1) All happy primes are happy numbers
(2) All happy numbers originate in California
Conc: All happy primes originate in California

Now, his little mishap with confusing happy numbers and happy primes somehow does not set Mr. JCB back at all, considering he somehow stumbled into an example using a happy prime. What does set him back is his misunderstanding of the phrase "come from" as the much stronger "currently holds residence in". Under JCB's view, nobody is ever allowed to reside in a state in which he or she is not born. To refute this obvious gaffe, Mr. JCB need not look any further than a mirror.

Furthermore I believe that his understanding of residence is wrongheaded as well. The property of 'being the number, x, of electoral votes a certain state, y, has', Vyx, is not a property indicating residency, and furthermore it is not singular, 'V7Connecticut ^ V7Oklahoma ^ V7Oregon' is not false, and if it were to become false it would be because of factors of the states, and not because the number seven cannot have the relation V to many states.

Thus i think it is obvious that JCB's baseless attempt to refute my view and derail the work of legitimate philosophers has been show to be foolhardy. I would suggest to Mr. JCB he try a little harder if he wishes to provide any worthwhile criticisms of this hypothesis.

(Sorry, no time to find useless links this time.)

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...have to take time out of my busy...

Post  John on Sun Nov 04, 2012 4:51 pm

Even better:

(1) Happy primes come from Happy numbers
(2) Happy numbers come from California
Conc: Happy primes come from California

JaSanberg has rightly pointed out that, in my haste to go to the gym, I examined his argument incorrectly. Still, despite his casuistry, there are several reasons why his initial point is at best misleading:

...he left out the most obvious and coherent of all. That position is that numbers do not exist as abstract concepts, but they do exist-- Numbers are concrete objects in the real world.

I mentioned this as a possible response by the mathematician, but I am sure JaSandberg was nodding off, so I will forgive this misrepresentation. Speaking of misrepresentations, however, this above quote, when combined with this response:

...not because the number seven cannot have the relation V...

Or the more damning definition of concrete:

Every number has some physical representation in the physical world. It is a fact that is overlooked by most people, as when you are not looking for it, it seems impossible that a street lamp is actually 1.5783 or that a brick might be the number 9. But once you've accepted this position it is quite reasonable.

seems to imply two objectionable consequences. First, that 'this is a fact overlooked by most people' seems fitting since it is false. Saying that 'once you accept a position it is quite reasonable' is spurious, though if JaSandburn considers what this might entail as a narrative description of why people accept statements like, "a street lamp is actually 1.5783," then he might see how vague and unfulfilling a term like 'reasonable,' is.

Second, no reasonable person would argue that numbers cannot hold relations, but the commitment to actual numbers is not entailed by use. Consider phlogiston. For some time, this was considered the cause of thermal exchange between objects, though this view is today considered false. Phlogiston, however, worked well within the developing system of thermodynamics, especially since experimental evidence supported its existence. Unfortunately, later research led to its downfall and ostracism as a concept. Because a model, say, arithmetic, is useful, does not entail that its objects, say, numbers, exist.

Now, let us take seriously for a moment, the idea that numbers are concrete in the common, more tenable, definition, namely, as spatiotemporal objects. Then, if there are such objects, one could certainly see how they could be referenced in theories. I mean, look, there is the number 1, and when you put it together with itself you get another number, 2. Superficially, these would be strange objects no? I mean, putting 1 thing together with itself resulting in something else is fine, but are there two such identical objects? How can this be? If anything, this would imply that there is a concrete 1 in space and time, and another 1 in space and time, and they are identical. But if they are identical, then they must occupy the same space and time, but this does not follow from the description. Not to be glib, but if there are two bushels of hay and they are identical, then they are the same bushel of hay. In short, there seems to be a violation of the law of identity entailed by concrete numbers, but this is, to be sure, beside the point (though consider adding to three, or four, or eight million...what trouble...U.S. students think they have trouble with math now?!).

Numbers are just the beginning though. A stronger response to the Oracle argument would entail that all abstract objects are, in truth, concrete. This includes father-of functions, set-theoretic notions of truth, semantic value, and mereological assumptions. But wait, mereological assumptions are considered abstract? I hear you think. Why certainly, and they also underline developed constructions of numbers. For instance, take a coke can and a table. Why is this two things instead of one? Mereological assumptions. More to the point, take this natural number 2. It is allegedly composed of 1 and 1. Bypassing the identity issues here, it seems that the now concrete function 'composed of' binds the formerly abstract numbers together to make a new thing, 2. But the mereological assumptions must exist concretely as well, so why look for numbers when we could be searching for those instead? The problem is, of course, that there is no established system for determining when we have 'found a proper mereological assumption' as opposed to the established system for determining when we have 'found the number 2.'
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