Posted by on Apr 17, 2013 in Blog, Numbers | 0 comments

Numbers: sets or successors?

When I was in high school, I played football. My position was defensive tackle, it’s a rough game; and eventually, I wound up with torn cartilage in my left knee. I had surgery for that when I was sixteen, and recovered quickly. When I got to university a year later, I was a philosophy major. I was reasonably good at math & science, but I didn’t find it very challenging; so philosophy for me. Between my first and second year in that, I picked up a summer job washing cars. All the up & down movement, bending over, and et cetera, re-injured my knee. So, off to hospital for more surgery. Then it turned out that I had an allergy to the betadine that they applied to sterilize the surgical site. When the nurse came to remove my bandages after the doctor had discharged me, all my skin came off. It was totally painless. The nurse looked at me with a shocked look on her face; I looked back sort of curious, I guess. I suppose she expected me to be screaming my head off; but I felt nothing. I guess the nerve endings went with the skin. The upshot was that they kept me in hospital for another 3 weeks to make sure I didn’t get an infection, since a lot of skin was gone. If you looked at that knee now, you wouldn’t imagine anything odd had happened. Of course, there are the two surgical scars; but the big patch of skin that took months to heal back, well, nothing to tell it had ever happened.

Anyway, between spending nearly a month in the hospital, and then being signed up for physiotherapy on my leg a couple of times a week, I didn’t find another summer job that year. So, instead of lying around completely idle, I decided to pick up some books from the university library to get ready for my next courses. Since I was taking some logic classes, I started off with Russell & Whitehead’s Principia Mathematica. Among other things, I followed that up with Wittgenstein’s Tractutus Logico-Philosophicus. A little light summer reading. I’d sit in the back yard and let the sun bake my purple knee back to health, per the doctor’s orders, sipping on lemonade and boning up on logic.

The aim of the Principia was to derive all of mathematics from pure logic. I believe that the current view is that it was a massive failure. Kurt Gödel was especially critical of the notation, the lack of formalism, and errors. Eventually, Gödel published his incompleteness theorems, which pretty much demolished the axiomatic approach that Russell & Whitehead had attempted. For his part, Wittgenstein first criticized the Principia for how it handled infinities, (infinite sets or lists), and later criticized a revised version (which had been modified to handle his criticisms) for not being able to handle large numbers effectively.

Principia derived the notion of number from that of set theory. In spite of being attacked by the likes of Gödel and Wittgenstein, this set theoretical approach was what they taught me in high school mathematics. In Principia, a set is a simply by-product of a true statement of the predicate calculus. For example, I might assert that the statement that “there exists at least one x such that the color of x is blue” is true. For the Principia, this tautologically implies that there is a set of blue objects, call it B and x\in B, meaning that x is a member of the set B.

Principia derived a set “1” in this manner. Here is how Principia derives the arithmetic for “1 + 1 = 2”:

Theorem 54.43

Theorem 54.43

You can read the statement as “it is true that for \alpha and \beta members of the set “1” and if the intersection of \alpha and \beta is the empty set (the symbol sort of looks like a capital lambda, but isn’t) then the set union of \alpha and \beta is a member of the set “2”.

If you think for a moment about the cardinality of a set, which is basically the number of its members, you are already using natural numbers to count. So that concept of set theory presupposes a set of natural numbers and a counting operation. In Principia, the set “1” is defined, more or less, from the predicate calculus proposition “there exists x such that x is one thing” which is a bit like the ontological proof of the Deity in which existence is assumed to behave like an attribute. If we think of a set as being like a bucket that contains objects, there is a strong distinction between saying ‘the bucket is empty’ versus saying ‘the bucket does not exist’. The set “1” in Principia might have an infinite number of members. Likewise, does the set “0” have an infinite number of members, that is, is it a bucket full of empty containers, or is it something that doesn’t exist? Go back to the theorem from Principia that is reproduced above and you’ll see that there is a condition on \alpha and \beta that their intersection is the null set. Clearly, the authors want to make sure they’re going to get a set of cardinality “2” in the union. The more you look at it, the more it seems to fall apart.

There is a view, stemming from Russell’s paradox, that this approach to identifying sets with definable collections (naïve set theory) is the heart of the problem. It is possible to define all kinds of self-contradictory properties, such as lists of all lists that do not include themselves that should include themselves, and so on. The fix comes in what is called Zermelo-Fränkel set theory with the axiom of choice (aka ZFC). In ZFC, there is a recursive definition of the natural numbers.

One begins with the empty set, {}, which presumably is a bucket with nothing in it. We then, recursively, put something in the bucket: n+1 = n\cap \{n\}. Hence “1” = {{}}, “2” = {{{}}}, and so on. The underlying idea is that of an initial condition and a successor. Whether the initial condition is “0” or “1” is a matter of dispute in the context of natural numbers; but that “2” is the successor of “1” is a matter of no dispute. The most natural and obvious criticism of this proposal is that it is just counting in the context of the symbols of set theory. To quote the author(s) of the Wikipedia article on set theory and number:

  1. Zero is defined to be the number of things satisfying a condition which is satisfied in no case. It is not clear that a great deal of progress has been made.
  2. It would be quite a challenge to enumerate the instances where Russell (or anyone else reading the definition out loud) refers to “an object” or “the class”, phrases which are incomprehensible if one does not know that the speaker is speaking of one thing and one thing only.
  3. The use of the concept of a relation, of any sort, presupposes the concept of two. For the idea of a relation is incomprehensible without the idea of two terms; that they must be two and only two.
  4. Wittgenstein’s “frills-tacked on comment”. It is not at all clear how one would interpret the definitions at hand if one could not count.

To my humble point of view, this notion of “0” and “1”, necessary to get the set-theoretic concept of number and counting going, is very closely related to the notion of state in classical physics. In classical physics, a system can be in one state only. A paradigmatic example might be a coin on a table-top that can be in either the “heads” state or the “tails” state. Now, in more modern physics, certain conditions of symmetry are allowed; for example, the condition in which the coin is tossed and still flying through the air has a symmetry between “heads” and “tails”. But this condition of symmetry requires a certain amount of energy, on the one hand, and is unstable, on the other. Sooner or later, the coin falls, the symmetry is broken, and one or the other condition arises. If we were tallying up “heads” and “tails” in an experiment, this is when we would add “1” to the appropriate column in our tally sheet.

“State” and “count” or “measure” are in many ways synonymous. We might count sheep, let’s say, or black sheep, or sheep in the west pasture or black sheep in the west pasture. But to count any of these kinds of sheep, we have to be able to either find them, or at a minimum, imagine finding them. I do not mean in the past-time of counting sheep to fall asleep, rather, I mean estimating the size of a large population where obtaining a complete count is impractical. In this, I am trying to draw a distinction between, say, counting the three sheep immediately in front of me versus counting all of the sheep in the universe. Even counting all of the sheep on the Earth is a problem fraught with potential error. I don’t want to count dead sheep or unborn sheep. But if the radius of the Earth is around 6400 km, a signal from the far side has to travel at least 6400 π, or roughly 20,000 km to reach me. A radio signal to give me a count of sheep from there will take almost 1/10 of a second to reach me, and that ignores any signaling protocols to reduce errors and guarantee message arrival and accuracy. And some sheep will die and some will be born in that amount of time. So, the state of being a live sheep is not a stable one for this count. It is rather like the symmetric condition of the coin in the air: it can and will fall and become “heads” or “tails”, but I can pick it up again and toss it another time. So it goes with the sheep: the unborn one is rather like the coin still in the air, it could become a live sheep any time now. The live sheep is rather like the coin on the table: it can be picked up any time now and tossed again too.

This brings us to the kind of state transitions that classical physics allows. We cannot have ambiguous state transitions; that is, we cannot allow a sequence in which one state might transition into two possible outcome states. And by time reversal, we cannot have the situation in which two possible preceding states both transition onto one following state. Paths of states must be clearly delineated. Classical physics does not allow for this idea of symmetry breaking. For the coin toss, the classical physicist would have to argue that, in principle, the final condition of the coin is predictable from the exact details of the toss. This could be true, but there are so many tunable parameters and so many errors in measurement that, for any toss, getting the answer right is (a) difficult and (b) pointless. Having a good model for a coin toss comes down to having a random number generator that yields two distinct states with equal probability.

One of my philosophy tutors from back in the day took this business of considering state into the example of a cat. We all know what a cat is, don’t we? It’s rather like my example of counting up sheep. But where is the boundary of the cat? What is part of the cat? What about the air it’s breathing? Is that part of the cat when the air flows into its lungs? Or when the oxygen gets into its blood-stream? When precisely does the air become part of the cat, if ever? What about the air mixed up in all of its hair. There is a lot of hair! Those air molecules might have been trapped in there for years now; they may have a better claim at being part of the cat than the air it’s breathing, which comes and goes. There are many questions about what’s part of the cat. Or part of us. Almost every molecule in our bodies is replaced every 7 years or so. Our energy store is even more limited, which is why we’ll starve in a relatively short time. And yet we maintain a notion of self decade after decade. My memories of seeing the skin peel of my knee are still “fresh” as it were, even though, apparently, whatever material part of me that information is encoded in has changed many times over. “I” remain continuous: a single thing: a “1”. There is just one of me.

This question of state in classical physics is a big problem in the transition to quantum mechanics and quantum field theory. The notion of state in quantum theory is very different than in classical physics. In classical physics, a system is in one and only one possible state. In quantum physics, a system is in a linear superposition of possible states, each of which has some probability, at least until a measurement is made that forces a specific condition. To give an example, say we have an electron which can have its spin up or down. If I take an electron and measure its spin, I might get up or I might get down. This seems rather like the state of the coin: I toss it and get heads or tails. Like the coin, if I measure the electron’s spin again, after measuring up, I’ll continue to get up. However, suppose I know check to see whether its spin is to the right or left. I will either get right or left; it’s a 50:50 deal, just like the previous test for up or down. If I measure right or left again, I’ll get whatever I got the first time. My measurement has frozen in the state I first found. But now, let’s say I go back and try to measure if the electron’s spin is up or down again. Now, it is once more a 50:50 shot as to whether I get up or down. My measurement of spin in the right-left axis has unfrozen the spin in the up-down axis. Similar tricks apply to photon polarization states.

The main point is that in quantum mechanics, the state of a quantum system is in a kind of limbo until a measurement is done. More elegantly put, perhaps, the state of the system is indeterminate until it is entangled with the state of a measurement apparatus. Typically, the state of a quantum system is given by a countable set of eigenvalues that correspond to a set of eigenvectors. The eigenvalues represent energy states and the eigenvectors represent the wave function for the system in the corresponding energy condition.

This is, arguably, a question that goes straight to the heart of what constitutes a proper concept of numbering in quantum theory versus that in classical mechanics. If integers, or reals, or vectors, or tensors, are the proper numbers for classical mechanics, then what is the right number for quantum mechanics?

Some very intelligent people have argued that number theory should properly precede set theory, and any attempt to derive numbers from sets has put the cart before the horse. I am rather inclined to agree. However, I am also struck by the strong relationship between the concept of state, classically, and the concept of number. This ties nicely with the “naïve” set theory model that a set is defined by some statement of the predicate calculus. Assuming that a statement of the predicate calculus has a truth value, then the classical mechanics notion of state tells us something very similar about a physical system; for example, it is true that the energy of the system is 10 Joules or that the speed is 299,792,458 meters per second. Classical mechanics is about true statements concerning reality. Quantum mechanics stumbles with notion: is it true that the electron spin is up or down? No, it depends: it always depends. And often enough, the answer is that the spin is both up and down. The law of the excluded middle completely fails in quantum mechanics. It is not that we do not know what the spin is: it is that the spin is both up and down equally. To bother at all with the truth value of the electron spin being up or down misses the point. We need a way to express the equality of the truth values of both up/down, right/left, back/front. The truth is all of the options, or as a physicist would say, the spectrum. Is the entire spectrum a number? More of this later…

Meanwhile, let’s begin with “0” and “1” (or {} & {{}} ) and see where we get.

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