I recently just finished reading H. O. Mounce’s “Wittgenstein’s Tractatus: An Introduction”. I had never read anything by Wittgenstein before, but had heard from many reputable sources that his thoughts are among the most influential in modern times, and that he is the father of the analytic movement that dominated the last half century. Actually, I bought Mounce’s book by accident, thinking it was the actual Tractatus Logico-Philosophicus, only finding out later that it was a sort of walkthrough, with explanations and commentary as to what the actual Tractatus meant. I think of it as a happy accident, as the Tractatus is known to be an extremely difficult read.
The main thing I took away from Mounce’s explanation was that Wittgenstein saw logic not as something that could say what was or was not true in the world, but could only help us form proper statements, which themselves would have to be empirically tested in order to be true or false. That is to say, logic is an invention, that we formulate in order to hopefully mirror the way the world works.
This reminded me of mathematics, and an argument (read: short exchange of words) I had with my philosophy of physics professor. We were discussing math, and he said that he believed it was something that was real in the world, and that we discovered it. Whereas I took the stance that mathematics was invented, and can be invented in any number of ways, and that we make it fit how we see the world. That is, I see math as a perfectly working system, whose axioms and rules we chose because they fit our observable world.
I can’t remember where I got this idea from, perhaps GEB, or What is Mathematics?, but the book I think about most when it comes to this idea is Brian Clegg’s “Infinity: The Quest to Think the Unthinkable“. In it he discusses the use of infinity with respect to calculus. It got me thinking about whether or not space and time are discreet or continuous (digital or analog). In order for me to explain myself, however, I’ll have to digress and talk about the definition of a derivative, and the use of limits.
Alright, let’s take a curve, like a parabola (which upside down is like the arc a rocket will form when fired (note: I’m reading Gravity’s Rainbow right now, so this example is quite apt)) which can be described by y = x^2. If you wanted to find the rate of change of y with respect to x (how y changes with a changing x), you could try and find the slope at that particular point. If you remember highschool math, you’ll know that a slope is described by ‘rise over run’, or: m = y2 – y1 / (x2 – x1). If you want to do this for a particular point, you’d want the difference between y2 and y1 (and x2 and x1) to be very small. In order to do so, we invent a variable that represents the veeeeeeeeery tiny, which the symbol Δ denotes when beside a variable. Thus Δx is the veeeeeeeeeery tiny in the xs. Ok, so let’s take the slope between two points, but the distance between these points we want to be veeeeeeeeeery tiny. So we’ll pick x (x1), and x+Δx (x2), and our ys will be (x)^2 for y1, and (x+Δx)^2 for y2. Thus our slope will be: m = y2 – y1 / (x2 – x1) = (x+Δx)^2 – x^2 / ((x+Δx) – x) = x^2 +2xΔx + Δx^2 – x^2 / Δx = 2xΔx + Δx^2 / Δx
I’m so sorry for this…but yeah, at this point, we can factor out Δx from the top, so we have: Δx(2x + Δx) / Δx
Now, since Δx are veeeeeery tiny, but not zero, we can cross out Δx from the top and bottom (something divided by itself is 1), leaving 2x + Δx. Alright, now, if we take Δx to be so small it doesn’t significantly add anything to 2x, we just drop it. So we’re left with 2x, which will be the slope at any given x you sub into it for y = x^2
Sorry for that (and for any mathematicians out there, I also apologize if I fucked that up, I honestly just did that from the top of my head), but I feel it was necessary (and fun! Right?! Huh?! Yeah!?) to make my point about limits, and infinitesimals being used for calculus. What I want to discuss is whether or not, in calculus, if veeeeeeeeery large numbers (integrals), or veeeeeeeery small numbers (derivatives, as my example provided), are actually inifinite or inifnitesimal, or are they just large enough or small enough? You may think that this isn’t important, but let me digress once again and discuss Zeno’s Paradox, which will ultimately lead me to explain how I feel the universe would make more sense if space and time were discreet (at one point there is the very smallest piece of space or time, that is, where a quiptometer, or a quiptosecond (hopefully those words are made up) would be the smallest units of space or time).
Zeno’s Paradox, if I remember it correctly, can best be described as follows. Say you are in a room, and you want to go from one wall to the other. In order to get to the other side, however, you must first get half-way. In order to get half-way, however, you must get half-way to half-way. In order to get there, however, you must first get half-way to that half-way. And so on, and so forth. The paradox is, since you can do this ad inifitum, and thus must travel an infinite space, how on earth do you ever get to the other side of the room?
There is also the same paradox as applied to time. The conclusion of this time paradox was made abundantly absurd when at a party I went to a long time ago, a friend of mine said that he “did not believe in time”. As he told the paradox, and concluded with his surprising statement, people began to argue with him. As this happened, he was interrupted by a phone call, to which he paused his audience and said “hold on a minute”, at which point another friend of mine jumped up and exclaimed “HA! Are you serious? Did anybody just catch that?”
Zeno’s paradox had a similar effect on myself (although I didn’t explicitly deny space and time). It just kind of festered in the back of my mind without any resolution. That is, until one day when, in a friend’s Cognitive Science class, the prof, in one fell swoop, dismissed Zeno’s paradox by saying that calculus had solved all that, with limits. At the time, I hadn’t read up much on calculus and infinity and limits, and so was very interested in his answer, but wanted to know how it worked. Thus, after I had read those books, I was left disappointed. I failed to see how calculus actually used real infinities, as opposed to the VERY large and VERY small. If the latter is the case, then Zeno’s paradox remains unresolved. So what then is the answer?
Actually, the way I resolve the paradox (I have no idea if this is correct), came from something I had heard before, sometime in first year physics. A prof was talking about time and space, and was digressing about whether or not it was continuous (can be continuously divided) or was discreet (had a smallest quantity). He was talking about the smallest unit of space, and the smallest unit of time, with which we can current measure things to. He also speculated as to what the smallest unit would be, and I believe he suggested planks constant (or h, to those who are familiar with quantum mechanics).
If this is true, if there is a smallest unit to space and time, then Zeno’s paradox becomes resolved because there is no infinite divisibility. You see, Zeno’s paradox relies on the assumption that space or time are infinitely divisible. However, if at one point you can no longer divide, then there is no longer a paradox: in order to get from point A to point B, you have to traverse a finite space (in a finite time).
Are there any current theories in physics that assume a discontinuous space and time? There is actually! There may in fact be many, but the one I am familiar with is Loop Quantum Gravity, a contender (along with string theory, which has recently fallen out of favour) for a theory of everything (combines relativity and quantum mechanics).
All this comes back to Wittgenstein, with logic and math being inventions, not actual measurers of the universe. Does calculus indeed use infinities? Does it just use large and small enough numbers in order to make it work? Does it matter? If space and time are continuous, then perhaps calculus, using infinities, does solve Zeno’s paradox (I’m not convinced). If, on the other hand, space and time are discontinuous, then calculus is not wrong, since it does not presuppose infinities (however, I do feel as though Zeno’s paradox is better resolved if the universe does turn out to be discreet.) Math is an invention, which we can alter or invent in order to mirror the measurable/testable universe.