Dear Ian Durham,
I guess no one told you, but the Queen of Hearts has spoken. There are 3.14159 names on the list today and to anyone who disputes that "off with their heads"!
Regards, Ed Unverricht
The Raven and the Writing Desk by Ian Durham
Hi Tejinder,
I am actually aware of Cantor's proof and have even taught it before. Last night I was reading Russell's heuristic argument. It seems that these proofs rely on the property of reflexivity. In thinking long and hard about it, I think my problem is that I have not been convinced yet by any proof of the reflexivity of the integers, deals, etc. And it also seems to me that compactness is, in a weird way, a type of infinity in a sense. Obviously this is counter to the orthodoxy but I suppose I am more like the March Hare than I might be willing to admit.
Cheers,
Ian
Dear Prof. Durham,
Thank you for this delightful dialogue. I really enjoyed it.
I would like to invite you to read my essay; it also deals with the relation between mathematics and physical reality.
Best,
Mohammed
My apologies Ian, for not realising you had something deeper in mind, beyond Cantor. Yes indeed we need a number theorist to opine on this :-)
I have enjoyed reading your essay, and its style :-)
Best,
Tejinder
Oh, no worries, Tejinder. But we do need a number theorist... :-)
Sorry I can't help, other than to note that reflexivity is another one of those features that puts holes in our proofs as well as our intuitions. Perhaps someone can develop a theory of partial cardinality... not that it would help much.....
Cheers - George
Hi Ian, this essay is really lovely. I think tangible mathematician, not the representational pseudonym Lewis Carroll would approve- If he was alive, Which he isn't....
I'm beginning to feel like I'm one of your characters.
Is the universe tangible? I think the visible universe is representational. Formed from received data but not out there touchable as the data takes time to arrive and a universe in motion continues in motion always becoming.So not as seen.
We can assume there is a tangible universe but it is only potentially so- as we cannot reach it to touch it, our probes only just leaving our solar system. So a bit like the potential infinity that might be counted but can't actually -unless you are a fictional talking dormouse! I wonder whether 'concrete' rather than 'tangible' may better capture the sense of being in some way more actual than abstract or representational without the necessity of being touchable.
I love, Quote: "Hatter: After all, just because I'm a character in a dialogue doesn't make me any less a part of the universe." Which gets us thinking about how different kinds of reality relate to each other. Certainly the ink on the page or pixels on the screen coding the dialogue are tangible but they are also representational and the characters represented almost 'come to life'and almost seem tangible but are despite wishful thinking only as tangibly real as Lewis Carroll deceased.
Great fun, best of luck. Georgina
Dear Ian Durham,
If you need a convincing number theorist I recommend Salviati. Of course, he as well as Euclid are presently kept for heretical or at least outdated.
Not just you might hopefully agree on that any cardinality in excess of the plausible distinction between an unbounded but discrete grid of numbers (a_0) and the endlessly divisible continuum (a_1) has not proved useful in science.
Do you think I'll stick with tangible reality? Instead I prefer conjecturing reality as something that fits to confirmed by experience and reasoning self-consistent basic relationships like causality.
Respectfully,
Eckard Blumschein
Hi Ian,
This is a very enjoyable, easy to read essay. But re
"HATTER: Neither is there anything particularly tangible about 'two' or 'ten. ' They are abstract concepts."
and
"MARCH HARE: And mathematics is representational? HATTER: Precisely. . .MARCH HARE: No. I simply don't buy it. I'll stick with tangible reality, thank you very much" :
If numbers are representational, i.e. if numbers represent physical reality, then you can't really say that numbers are "abstract concepts". What "tangible reality" does a number represent?
I contend in my essay (Reality is MORE than what Maths can Represent) that numbers MUST represent fundamental physical structures. And, though it's seemingly not a complete solution to the number "problem", I contend in my essay that a natural or a rational number must represent a ratio: a "relationship between information categories" where the category in effect cancels out.
Cheers,
Lorraine
Thank you for the kind words, Georgina. I am glad you enjoyed the essay! Indeed, how do different realities relate to one another? That is an intriguing question.
Hi Lorraine,
The number 3 can represent many tangible things: 3 donuts, 3 tortoises, 3 coins. In fact that is precisely the point of the abstraction. Russell, extending the ideas of Frege, essentially says that numbers are similarity classes. So when we say "there are three coins in my pocket" we are asserting that the objects in my pocket share some (possibly vague) similarity. We could just as easily say "there are three things in my pocket" and those three things might be a coin, a candy wrapper, and a key. While they are not terribly similar, they do share the common fact that they are made of ordinary matter and so we may classify them as such.
Conversely, I can't say "there are three airs in my pocket" (referring to the air we breathe). I can't "count" air in the sense of that sentence. I could count air _molecules_, but not "the air". As Russell points out, these ideas are deeply connected to the language we use to express them.
Ian
I must say, Ian, that the only essay I enjoyed as much as yours in this contest was the one by Tommaso Bolognesi,and have voted accordingly. Though I am still confused as to how all these mathematical madness gives us such accurate models of the world.
Please take the time to read and vote on my essay:
http://fqxi.org/community/forum/topic/2391
Best of luck!
Rick Searle
Thanks Rick! I think we're all confused about that. :-)
Rick and Ian,
"Mathematical madness" and "we're all confused about that. :-)"? Sounds like emotions of freshmen students who were confronted with a shut up and calculate attitude even in the simple case of using the definitely not mysterious complex calculus.
Maybe, Wigner intended bringing irrationality into matters that seemingly evade common sense? I noticed that he almost adored J. v. Neumann who on his part admired v. Békésy for his held for more genuine nobility. Irrationality is a basis for belief. Suppressed doubts in the correctness of their believe might have affected G. Cantor, Hausdorff, Gödel, Grotendieck and others. Cantor died in a madhouse, others behaved otherwise abnormal.
Incidentally, to those who don't understand my hint, in a fictitious dialog Salviati was used to utter Galileo Galilei's still compelling reasoning: "The relations smaller than, equal to, and larger than are not applicable on infinite quantities, only on finite ones".
Eckard
[deleted]
But Ian!
Re "3 donuts, 3 tortoises, 3 coins": You are talking about subjective structures/relationships that exist in your brain, which you can represent by the written symbol "3" or the spoken word "three" etc. There is nothing abstract about what exists in your brain, which you subjectively experience.
The point I was trying to make is: what is the reality behind quantity in FUNDAMENTAL physical reality?!! What "tangible reality" does a number represent? Surely, we've got to stop always looking to an abstract platonic realm to solve every difficulty?
I contend in my essay (Reality is MORE than what Maths can Represent) that numbers MUST represent fundamental physical relationships. I contend in my essay that a natural or a rational number must represent a ratio: a "relationship between information categories" where the category in effect cancels out.
Cheers
Lorraine
Above post was from me, Lorraine Ford.
Dear Ian,
Good essay. It shows the distinction between maths (the representational) and physics (the tangible) at an elementary level. Can you comment on the quantum reality? If one agrees that quantum measurements are contextual the quantum world is tangible, isn'it?
The cardinality of the Monster Group M (big but not infinite) is representational at the moment but it can become tangible (will be effectively counted with the computer), the cardinality of most sporadic groups became tangible.
All the best,
Michel
Hi Lorraine,
I think we're talking past each other here. I absolutely agree that we need to get away from Platonism, at least in physics. Several of my other FQXi essays have touched on this topic in one way or another. But I think the question of what tangible reality a number represents is a concept that really has little to do with number itself, with the exception that, as Russell pointed out, it is a way of creating a similarity class.
Think about it this way. Imagine a toy universe that only has two objects in it. What purpose would the concept of number serve in such a universe?
Ian
Hi Ian,
In the toy 2 particle universe you describe, only the following information can exist (from the point of view of each object):
1. Categories of information (about "itself" and the other object) e.g. mass or a more fundamental category of information;
2. Information interrelationships;
3. Quantity information associated with the information categories.
A physicist could represent this information and these information relationships with law-of-nature mathematical-type equations; and the quantities would be represented by numbers.
It's clear that reality utilizes "categories of information" like mass, momentum, charge etc. And some of these categories are not fundamental, but are in effect built out of relationships between existing, more fundamental, information categories. Seemingly, at least one information category must be considered to be a "first principle" of reality.
Similarly, it is clear that reality utilizes information interrelationships. We attempt to represent such interrelationships with symbols like "= + x", and at least some of these interrelationships must also be considered to be "first principles" of reality.
When it comes to quantity, which we represent with numbers, we must also assume that there is an underlying simplicity. "Quantity" describes something that really exists in fundamental reality.
But the objects in fundamental reality can make no distinctions relating to number of objects: there can potentially only be distinctions made relating to quantity. The reason for this is made clear by mathematician Louis H. Kauffman's article "What is a Number?":
"Two classes are said to be similar if there is a one-to-one correspondence between the members of the one class and the members of the other class. . . [But] How do you know to take a given form as foreground against which to make the comparisons?. . . None of these questions can be addressed by a formal system alone. The ability [to] perform the relations indicated by these questions is a prerequisite to being able to make any formal system at all."
Correspondence and comparison are complex many-step operations: clearly there can be no such ur-counting behind-the-scenes at the level of fundamental reality. Classes and sets have a hidden underlying complexity: sets have a complicated underlying infrastructure of axioms, defined symbols, defined equivalence relations, ordering etc. If one assumes that some sort of simplicity underlies physical reality, one must conclude that the quantities found to exist in fundamental physical reality cannot be modelled by the sets that represent the properties of number systems.
On the other hand, a number seen as a ratio seems to be possible for fundamental reality. To give another toy example: assuming a toy law-of-nature relationship could be represented as "a+bc =d", then a (rational) number relationship could be represented as "(a+a+a)/a" . I'm saying that the substance of the reality underlying quantity could be as simple as the reality underlying laws-of-nature.
(I attempt to explain in my essay how the complex numbers, algebraic irrational numbers, and non-algebraic numbers found in nature might be seen in the light of the above way of looking at numbers.)
Cheers,
Lorraine