Saturday, December 10, 2011
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About Me
- Kirby Olson
- Upstate, NY
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37 comments:
Ginormous.
Gihugic sounds like some sort of chemical, or perhaps an Asian form of exercise.
Ginormous just sounds big.
I agree with Brett, but I have a different, and two part, analysis.
The first is that I've heard "ginormous" a lot, especially from 20-somethings. I've never heard "gihugic." This general sense of relative usage is further confirmed by the dictionary program on my computer, which recognizes (and gives an appropriate gloss for) ginormous, but which flags gihugic as a misspelling.
The second is that both words are portmanteaus of "gigantic" with "enormous" and "huge" respectively. And "enormous" is bigger than "huge," so it is reasonable to assume that the intensified variants of each at question here stand in the same relative relationship.
I looked up, "Is enormous bigger than huge?" just for fun, because I had never heard this relative sizing. I found lots of people weighing in on this topic, among which was this amusing little statement:
"Go into Outlook 2003 and sort by size. Look at the sizes: Tiny, Small, Medium, Large, Very Large, Huge, and Enormous. At what point did enormous become larger than huge? Why not Gigantic? Ginormous? Probably either decided by one bored dev with a deadline or by a gaggle of program managers over the course of nine months. Decisions like this aren't made any other way."
I don't know. I don't really use words like gihugic or ginormous. Someone said that gigundo is a big one, now. They have giant and super giant stars. To some extent this reminds me of the debates surrounding various kinds of infinity. I heard someone say that the set containing all odd numbers is actually larger than the set containing all numbers, but the logic escaped me. I'm sure the logic was finely worked out, but it went over my head.
I think it's a misnomer. The prefix is giga not gi, unless we're talking giraffes.
Kirby,
To some extent this reminds me of the debates surrounding various kinds of infinity. I heard someone say that the set containing all odd numbers is actually larger than the set containing all numbers, but the logic escaped me.
I assume that by "number" you mean "natural number," i.e., non-negative whole numbers.
The set of all natural numbers is the same size as the set of odd natural numbers, because they can be put into a 1-1 correspondence. It's that simple.
infinity just means...you can always add an 1 to it. Not the Cantorian dreams that Doc Stu and his cronies envision. Who was the poetical who yapped "no ideas but in things"? Sorta like dat.
what ever happened to gargantuan
a perfectly good superlative
jh,
what ever happened to gargantuan
a perfectly good superlative
True enough, but one of the cool thing about language is that it is being continually reinvented. We may have to deal with the past, but there's no reason not to improve on it. Things get bigger all the time (consider, e.g., waistlines), so we need new and bigger words to describe them.
How about gargigantuan?
Stu's right we constantly have to refresh the vocabulary andinvent new ways to say things. Big and small are relative, anyways. To an ant my foot is a colossal cathedral, and if my foot lands on the ant, goodbye ant. To an ant this town is seemingly endlessly. But to Gargantua, there's not enough food in this town for brunch.
Size is relative. It can be objective as when we say that the New Jerusalem will be 1300 square miles. That seems to be a clear enough set of parameters. But we also always want to go into the subjective, and say that something is extraordinary! or majestic! which are things that can't be measured, but might be shared among several. The notion of the sublime comes into play in John's description of the New Jerusalem.
He strains credulity, but that's part of the amazement of it. It's just so big! And it shines from within because of its beauty.
It reminds me of the detailed account of the ark of the covenant. The Biblcial writers spent a lot of time on objective description but they were aiming at the sublime. Believability is created through the objective measurements. But the sublime is made up of the aggregate of the parts, and the sheer bigness.
How big is God's face when Moses sees it? And when we see it, will it be the same size as when Moses saw it, that is, if we see it? (some of us may only see the devil.)
Kirby wrote,
Stu's right
Wow. Can I frame it? ;-)
Christendom as SK would say
idle chitchat, comedy, irony 24-7. Kirbyism
J,
This is hardly the venue for another logic battle, but...
infinity just means...you can always add an 1 to it.
Yes, but I'd opt for Dedekind's definition -- a set is infinite if it can be put into a 1-1 correspondence with a proper subset of itself. The two notions are connected, of course, via his rigidity theorem (any two models of 0, S, and second-order induction are isomorphic).
But the question here isn't whether or not infinites can be reified in mathematics, it's about the theory of relative size -- specifically, when does it make sense to say that two distinct infinite sets have the same size. The notion of a 1-1 correspondence seems like a mathematically natural generalization of a the notion of "same size" from the finite.
Not the Cantorian dreams that Doc Stu and his cronies envision.
There's nothing wrong with Cantor's theory, based on simple correspondences. The big leap, of course, is that any two sets are commensurable under this theory. This follows from AC via a standard Zornification, but it might be weaker. I'm not sure.
But even if you want to stick with relatively concrete notions, you either have to deal with the notions that not all infinities are the same (specifically, there are more points on the continuum than there are natural numbers), or you have to engage in Broweresque fuzzy thinking over the nature of the continuum.
Who was the poetical who yapped "no ideas but in things"? Sorta like dat.
William Carlos Williams, something I've learned by following this blog. It's a strange thing for a logical/computer scientist to know. But I'd argue that this quote ultimately argues against your position. If you're skeptical about higher infinities, that's your right. But if the continuum isn't a "mathematical thing" to you, then your mathematics is impoverished.
I love to listen to Stu's mathematical side, and only wish I had the chops to engage with it in some way. I do understand the one to one correspondence paradigm, and also the +1 concept. There used to be a mathematician here as provost and I would occasionally loft him questions, and he would explain things in such a clear-minded way. Plus, I do try to read books that hang at the popular edge of mathematics and thus are within easy reach. Quite charming stuff!
Well at any time in the bijectional series sum up the natural numbers and their pairs in the odd numbers and the NN much larger magnitude at least.
J,
Well at any time in the bijectional series sum up the natural numbers and their pairs in the odd numbers and the NN much larger magnitude at least.
I'm sure that I don't follow.
For those who are trying to follow along, we're considering the relative sizes of two sets: the set N of natural numbers {0,1,2,3,...} and the set O of odd natural numbers {1,3,5,7,...}. Intuitively, O has half as many elements as N does, because it's "missing every other element." But intuition can break down in dealing with the infinite. Two infinite sets have the same size if they can be put into a 1-1 correspondence, which we can do with N and O as follows:
0 in N corresponds to 1 in O;
1 in N corresponds to 3 in O;
2 in N corresponds to 5 in 0; and in general
n in N corresponds to 2n+1 in O.
Indeed, any two unbounded subsets of the natural numbers
a_0 < a_1 < a_2 ... and
b_0 < b_1 < b_2 < ...
can be put into a 1-1 correspondence by the simple expedient of having
a_n correspond to b_n. This is essentially the argument of Dedekind's rigidity theorem.
Now, you're suggesting that if we sum up the natural numbers and compare this with the sum of the corresponding odd natural numbers, that the natural numbers will have a larger magnitude. This isn't relevant to the issue of the relative sizes of the sets, and as I interpret the statement, it's not even true. For example, if we sum up the first three natural numbers in this correspondence, we get 0+1+2 = 3, whereas if we sum up the first three odd numbers, we get 1+3+5 = 9, and 9 > 3.
Well, I'm happy we've established this. I have to say that J's position strikes me as the one that most Americans would intuitively back, but it just shows how out of touch we all are wth regard to higher mathematical reasoning. It's taken me years to follow the logic of the various infinities, but I am beginning to see how Cantor's logic -- working from the ground up, as opposed to working from the overall set theory and working down, is superior to my original reasoning on this. Perhaps someone will overturn this, and convince even important mathematicians of another way of looking at it, but I am sure if that person arrives, it will be neither myself nor J.
zero is an abstraction
Well, depending on how you set it up, a set of exponentials would be larger (if totaled up) numerically than just integers. That isn't the point. The incommensurateness of the sets is--they can't both be infinite. Just that the process seems that way (tho not in physical reality). Ie, it's just different ways of dividing up...nothing really. The sets have no actual existence (the operation, n + 1 does in some conceptual sense). There are quite a few who have taken on Cantor (Quine for one), and it's a daunting task unless one realizes.."no ideas in things" (even as posit). Mathematics is a tool. Not the mind of allah.
No, Stu it's the point on the cardinality (IIRC. I don't claim to be a set theorist, grazi a Apollo). There are 10 numbers from say Set A...1 to 10. There are only 5 odd numbers in Set B from 1 to 10. 10 > 5. And added-- Set A sums to 55. Set B to 25. 55 > 25.
I wonder if JH's position is accurate for all numbers. Numbers don't exist int he real world and without people I don't think would exist at all. So I think they are all abstractions. Maybe ducks can count to a degree? Do animals have math? Are dogs counting when they shepherd a thousand sheep?
More than a few mathematicians objected to Cantor's ideas:Kronecker believed that Cantor's paper was meaningless, since it proved results about mathematical objects which Kronecker believed did not exist.
That's it. Proofs about things that do not exist don't mean much (or anything). QED. (Im jesting sort of Stu but perhaps you can play devil's advocate for a few seconds and tell us how does one argue against the
counter-intuitive ideas of Cantor's infinities of infinities)
Stu, this is a good question that J raises. If there is a method to oppose Cantor's madness, where would it lie. It seems that he has to be wrong somehow. Has there even been any fruitful opposition to his one to one idea? I can imagine Quine going at it, but I doubt if he had the mathematical skills. I wouldn't know about any other attempts, and salute J for bringing Quine to mind.
I have no idea how you'd set about refuting Cantor since his argument is at once extremely simple and diabolically complex -- a work of pure and somewhat sick genius.
I do know how I'd go about refuting WCW's notion that there are no ideas but in things. I'd use Kant to indicate that there is no perception of things without ideas.
But WCW is quite easy to crack if you have even a tiny background in philosophy. WCW had some native intelligence but wasn't trained to think philosophically, and he was basically an imagist. He did have a concrete grounding due to his medical training. But I don't think his ideas can be defended philosophically. His economics are just plain weird.
But in medicine I think he was probably sound enough.
Mathematics is such a daunting field. Cantor's brilliant maneuver is so simple that I would have no idea how to refute it and yet it has something of the same bizarre genius without a trace of common sense that you see among the likes of Zeno, and his idea that movement is impossible, which he then sets about to prove through halving everything, and saying that therefore nobody ever arrives.
I love this kind of logic, and yet it's obviously somehow wrong.
Well, that's not exactly what Im saying. Computer science makes use of Cantorian ideas--including ordered-pairs and diagonalization (in some form--e.g., arrays. even a spreadsheet in a sense sort of basic ST--tho' others might just call that functional in a sense). It has some application--ie, it works in some limited contexts. But in a grand, philosophical sense, we should object to the idea of uncountable, unknowable infinite sets and so forth--the sets aren't actual entities in a platonic abode, though academic set theorists seem to think so (We leave the proof that the set of real numbers is not numerable as an exercise).
I've been at a meeting all evening, hence my delayed reply.
You guys are asking deep and controversial questions about the philosophy of mathematics, some of which have been around since at least Plato.
Is mathematics is invented or discovered? My experience in this is mixed, and so my personal philosophy is heterodox: I believe that definitions are invented, but theorems are discovered. Complicating this is the notion of a construction, which has aspects of both, and so often feels invented in part, and discovered in part. Whether such experiential data has any philosophical utility is a question that I'll leave to others.
So I agree with jh. As a defined concept, zero is an abstraction, as is one, two, etc. But these are abstractions from real world phenomenon, so abstract and invented though they may be, they are concepts that can be taught and shared, and their properties can be discovered and appreciated.
J is pushing a theory of size for subsets of the natural numbers based on density (this is somewhat obscured by his discussion of magnitude-density, but magnitude is a red herring asymptotically -- figuring out why is a nontrivial but satisfying exercise). Putting together a satisfactory theory of size for subsets of the natural numbers is a complex topic, and to put it plainly, there are quite a number of ways to do it, including (1) point-mass measures, (2) Cantorian size, (3) density, and (4) nonprinciple ultrafilters.
J approaches the question (perhaps in polemical jest) as if there's a right choice to be made here, and the Cantorian theory isn't it. I don't see it that way. All of these definitions are mathematically well-formed, all are potentially mathematically interesting. The question really boils down to which definitions have the most fruitful and satisfying consequences. E.g., only the Cantorian theory is invariant under permutations of the set of natural numbers. And all of the theories *except* density are able to assign a notion of size to all subsets of the natural numbers.
Historically, the Cantorian theory had more beautiful consequences, the philosophical reservations of Brouwer, Weil, et. al., notwithstanding, and this is why it has tended to dominate.
More recently, Szemeredi's theorem has invigorated interest in density. Briefly, Szemeredi proved that sets with non-zero density have arbitrarily long arithmetic sequences, a result that had been conjectured by Erdös and Turán many years before. This has fruitful combinatorial consequences, and is the subject of vigorous ongoing research.
Size and density are good ways to approach the problem of Cantorian sets I would think. Both are quite abstract. Cantor's approach of going from the bottom up is neat because it's susceptible to proof. It does remind me somewhat of Zeno's move to halve each motion until no complete motion is available.
Aristotle overcame that move by simply arguing that the whole move is available, as everyone knows in everyday life. We arrive places.
A similar common sense kind of move might again not be so susceptible to proof, because math is a separate sphere, and yet, as Stu says, it is derived from our daily experience.
If there are two natural numbers for every odd or even, you'd think yuo'd arrive at a set that was twice as big. But you could easily reverse that, and say, but there are two odds for every natural number, and show that, too.
But at least in the first twenty numbers there are twice as many naturals as either odd or even. You'd think that would suffice.
I understand that these arguments about various sizes of infinity get to be so big and unwieldy that cleverness is probably given an inappropriate amount of weight and simple commonsense (which, after all, can be wrong) is therefore dismissed as missing the point.
But Zeno was already working on these kinds of infinity and demonstrating it via simple down to the ground proofs.
It's funny how Cantor seems to have done the same thing and stumped everyone all over again.
Platonist Im not yet I don't think Cantor ideas are traditionally platonic, per se, except in a somewhat trivial sense. The greeks post-pythagoras craved axioms and geometric order. Perhaps Doc Stu can correct me, but they deprecated mere integers and arithmetic --Euclid bisects the unit circle in various ways and rarely depends on mere arithmetic, and really the irrational aspects of numerical series (Pi...) were troublesome. The geometric solids are abstractions from space,IMO. (integers too perhaps but there's a longer story to tell). The greeks were pragmatists in a sense as well (tho not usually read as such): mathematics served a purpose. geometry works for buildings, farms, ships, weapons etc.
J,
The ancient Greeks had mathematical interests beyond geometry (although it was often an anchor point). You'll recall Euclid's algorithm for computing GCDs, which argues for a fairly sophisticated understanding of number theory, primes, the difficulty of factoring, etc. And then there's the Pythagorean theorem, which among other things justifies the use of 3-4-5 triangles (which might be physical artifacts made out of rope) as means of creating right angles. They also had a serious interest in trigonometry, and knew how to compute various trigonometric functions, which made it possible to apply trigonometry to building, surveying, navigation, etc.
And don't forget about the Antikyra mechanism.
I've always thought of it, philosophically, as different magnitudes (or kinds or densities--though I don't think that's satisfactory philosophically) of infinities.
That is
a line is infinite in one direction, a plane is infinite in two directions, and space is infinite in all directions.
These infinities are clearly (though not, perhaps, mathematically) different and it is useful to think of them as different.
One can then add at least the fourth dimension (which, already infinite in all directions is therefore in all possible positions of all directions) and so forth but then things get all tesseracty and I don't really want any of us folding the universe this evening.
GM, that's a nifty way to explain the difference by doing the four dimensions! Each one is infinite, but some are seemingly larger still than the others. Nice idea. I looked up tesseract which was also a neat addition to my vocabulary. Thank you for this. I didn't know what you mean by folding the universe at the end but it was a charming flourish to your preceding composition!
GM,
a line is infinite in one direction, a plane is infinite in two directions, and space is infinite in all directions.
These infinities are clearly (though not, perhaps, mathematically) different and it is useful to think of them as different.
This is a useful and fruitful intuition. The resulting mathematics go in a few surprising directions.
The first, and arguably most surprising result, is due to Sierpinski, who showed how to construct a continuous(!) function from [0,1] (the unit interval) onto [0,1] x [0,1] (the unit square). The important word in the preceding section is "onto," which means that the range of the function is the entire square. This sort of function is commonly known as a "space filling curve," and a consequence of its existence is that the unit interval and the unit square can be put into a 1-1 correspondence, i.e., from the Cantorian point of view, they have the same number of points.
The story does not end here. For there have been many mathematicians who've felt as you do -- that the Cantorian equivalence obscures an important sense in which the line is different than the plane. And they've succeeded in providing definitions that enable exploration of the difference, but through topology (which deals with the *structure* of mathematical spaces) rather than through set theory. The result is various theories of dimension, and there are many, but they all share the constraint that R^n (n-dimensional real space) is always shown to have dimension n.
But the story does not even end here. There are subsets of the plane, e.g., that seem to be "bigger than a line, but smaller than the plane" from a dimensional intuition. Various fractals, e.g., have this character. This resulted in a the Hausdorff-Besicovitch dimension, and sets of dimension 1.5.
There's a nice page Wiki: List of fractals by Hausdorff dimension that contains a number of (often beautiful) examples.
If it is not perfectly clear, my position is that no yes/no suffices as an answer to the question, "is the line of the same size as the plane?," because the truth is, "in some senses yes, and in other senses no." It is useful to have both.
BTW, IIRC, string theorists posit the existence of an 11-dimensional reality, in which all but four of the dimensions (1 temporal and 3 spacial) are "folded up." I don't actually know any string theory, so I can't tell you what is gained by positing 7 folded up dimensions.
I'm very glad that even Stu admitted that there's something to GM's approach to the problem of various sizes of infinity.
GM, this is an amazingly clear illustration. I salute you for it.
As for string theory, I watched Brian Greene's PBS special on it a few years ago. I loved it. Then, last summer or the summer before that, I went to a conference that Greene had helped to put together on Science and education for young people. Greene lives near here in the summers. I was going down on the elevator, and he got on. It was just the two of us in an elevator. I couldn't say a word. I could have said, "I live near you!" Or, "String theory!" But I looked at my shoes.
I'm very glad that even Stu admitted that there's something to GM's approach to the problem of various sizes of infinity.
GM, this is an amazingly clear illustration. I salute you for it.
Kirby,
I'm very glad that even Stu admitted that there's something to GM's approach to the problem of various sizes of infinity.
This is materially wrong and prejudicial.
Let's start with "even." Yes, I debate vigorously, but I've never had a problem admitting when an opponent has a point. But I don't see GM (or even J) as an opponent in this discussion, so the whole framing that "even" suggests, which is of a debate and contention rather than a discussion and mutual exploration, is fundamentally misguided.
Next, "admit." I never made a claim that Cantor's theory was the only way to compare various sets, only that it was a sensible approach (contrary to J's dismissal of it). Indeed, I pointed out three other approaches to the question of sizing subsets of the natural numbers, and I'm happy to talk about dimension theory as well as a way of sizing geometric objects.
Scratch "even," and replace "admitted" with "affirmed," and you have a statement that is accurate and tends to advance the conversation in a productive direction.
There are different sizes of infinity only to those who start with an axiom of infinity ,and believe, nearly pentecostal-like, that unobservable, dimensionless entities (like sets, not to say uncountable sets) can exist. Many traditional mathematicians would disagree (as would most engineers).
A bridge-builder uses equations--load bearing, etc-- when planning his bridge, and applies them to the intended materials. Now he'd have a poorly designed bridge just using steel and no equations, like integrals. But he wouldn't have a bridge at all just using integrals.
Stu, I only meant that, as a PROFESSIONAL, it was interesting that EVEN you saw that GM's approach had merit. I just meant that you raise the bar further than the rest of us (I admit that my math skills are by the weakest of any of this group), whereas while J's skills in this area are not up to your level they are certainly well above mine, and Brett's, and probably that of JH, Craig, JADL, Emmy, Dim Lamp, etc.
But with great math skills often comes paranoia. We saw this with the beautiful mind movie. However, in this case, I meant this only as a compliment, but the compliment was pretty much directed at GM (but which depended on his passing by your standards, which I should think are quite high).
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