Tuesday, June 03, 2014

Sorry May

Okay, so this picture illustrates why I am not a Tegmarkian. Tegmark, if you don't know, is a clever cosmologist at MIT who's put forward (a book on) the thesis that mathematics is the ultimate reality, and that all mathematics is in fact a kind of reality - that there is a mathematical multiverse, which we know exists on account of the mathematics existing.

So I don't buy this. I'm diametrically opposed to this idea. Not opposed, really - I don't care too much, but I am opposed in that I believe the complete opposite. Mathematics - and physics as a subset of mathematics - is an artifact of the human mind, that's all it is. The fact that the world exists in some form is curious, although it seems incoherent to me that we can actually know anything about its true nature - but to suppose that its true nature is mathematics seems so backwards that I just wanted to write some things down.

I get where he's coming from. The world does exist, there is a reality, and it is somehow regular and consistent - it has properties that repeat or sustain, and why should it? Its continuities and discontinuities are all so numerically describable, and why should they be? And the most basic elements that we know to exist - photons, quarks, magnetic fields - seem to be perfectly and completely described as systems of numbers. And why should this be?

My mind seems to have taken the easy way out, because it just screams: but numbers and math are things that human minds *do*! They describe the world because the brain is a description machine, that's what it *does*! If the curious thing is that the description is so perfect and complete, then I have two responses - the space of possible descriptions that the mind can form is so vast, so impossibly vast, that it would be surprising if we could *not* find consistent systems of description for the world; and no description of the world is by any means *complete*.

The completion point is worth going on about. The scope and complexity of the natural world is impossible to comprehend. It's absolutely impossible to describe it all - and I'm saying this as a scientist with full faith in science as an endeavor for helping us to understand the world. We might choose some very narrow sliver of reality and subject it to intensive study, and then, there, we can describe it in such detail that we feel that it's okay to say we've basically got it all down. But that's it - those little, tiny, infinitesimally small splinters, and we think we have a complete description? What we have is a consistent system - mathematical physics - that can be used to describe anything we come across, but each description will be new, different, from what has been seen before.

So no description is complete. Okay, maybe that's a straw man, but I don't think so. Tegmark wants to claim that not only is physics a (potentially) complete description of our reality - or no, not a description, but *the thing itself* - but that realities we haven't yet encountered, i.e. realities *outside our reality* are contained within it. He likes the example of the discovery of Neptune. Astronomers had noted disturbances in the orbit of Uranus, and finally realized that there must be another planet even further out - they realized this mathematically, in such detail that they knew where to point their telescopes to find Neptune, and they did so, successfully.

Tegmark wants to use this example to imply that mathematics is a kind of tapestry containing all reality, and that by following it out from what was known, an *entire planet* was discovered, first in the mathematics, and only later by human senses. But this doesn't prove any kind of point about the reality of mathematics, and it's not even true, strictly, that Neptune was first discovered in a mathematical form. It was first discovered in the form of its gravitational influence, which affected Uranus. It's just that at first, astronomers didn't understand what they were seeing - they had to *do some math* in order to understand. But the data were all there - the measurements of Neptune in the flesh were there already, before Galle saw it with his own eyes (and others had seen it before, all the way back to Galileo, albeit not knowing what they were looking at).

The point here is that, really, new knowledge about the world can only come from new data about the world. Mathematics based on reality that has been observed - i.e. physics - can then tell you how to understand those data, but it is only that, a tool, an activity of the human observers. It doesn't exist outside of human endeavor. I am dead set in this opinion.

Anyways, so I basically had that conversation with myself last night on my walk home, and then I made that figure. It should be self-explanatory, but just in case: the biggest circle, the purple one, is the realm of all possible human thought. The circles within are not to any idea of scale, of course. There are many domains of human thought,and the next two that I've outlined are descriptions and axiomatic systems. Both of these I mean in the broadest sense you can imagine.  Physics falls within the realm of axiomatic systems of description, or it should (Hilbert's sixth problem). Within axiomatic systems you have consistent axiomatic systems, which should contain a correct physics, if it exists - i.e. if the Standard Model and General Relativity could be united. Taken as separate systems, I think that each of these theories alone counts as a consistent system, but together, so far, they do not.

Tegmark's reality is the domain of consistent axiomatic systems of description, of which our physics is (presumably) just a tiny part. Any other consistent system of physics would also fall in this domain, and Tegmark believes that each of these systems must also correspond to its own universe, just as our physics corresponds to ours. I think it's a fantastic idea, which I might illustrate by putting a big 'fantasy' circle somewhere in there, in between human thought and physics.

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