Mathematics is Metaphor

From Shut up and calculate:

I advocate an extreme “shut-up-and-calculate” approach to physics, where our external physical reality is assumed to be purely mathematical. This brief essay motivates this “it’s all just equations” assumption and discusses its implications.

Yet our ability to answer other questions has surpassed earlier generations’ wildest expectations: Newton would have been amazed to know that we would one day measure the age of our universe to an accuracy of 1 per cent, and comprehend the microworld well enough to make an iPhone.

There is a dangerous - and possibly naive - assumption in the above statement. We have models from which we can guess the age of the universe, but the basis for those models could be quite wrong. The models are based on what measurements and observations we can make from at most an insignificant cosmological distance from our home planet. The portion of the universe we have minutely explored and measured is fantastically small.

That we can come up with theories that seem to apply to the greater universe is - frankly - astonishing.

Still based on what we know the popular theories of the universe do seem to work. Good science is to go with what works. Good scientists should also keep in mind the limits of our present knowledge. Good scientists should be wary of our present assumptions, and question those same assumptions from time to time. Good scientists should be aware of the history of science - where further exploration of the real world (through experiment) have often forced changes in then-current theories.

Theory is a map of reality. Mathematics applied through theory is a means to read that map. But the map is not the territory. Maps are abstractions - they leave things out. If the bits left out do not matter for a particular usage, then the map is useful. But always remember - the map is not the territory.

Someday we may have a map of reality so detailed as to be indistinguishable from reality. Perhaps reality is sufficiently simple that a bunch of barely-evolved primates can cook up an accurate map. Somehow I just cannot come up with enough arrogance to assume that we are there already.

Over time I have worked with a lot of very smart people on fairly complex constructions. To get a grip on the both the people and the constructions, I have found much use for what I call “outer metrics”. Direct measurements of people or constructions may be practically impossible, but there are usually aspects you can measure readily, and that are indirectly but strongly related to what you want to measure. The linkage is probable - by no means certain - but useful.

The current state of Physics has me … wary. The “outer metrics” are telling me that the upper reaches of theory may be on the wrong track. Theory may be on the modern equivalent to the old “How many angels can dance…” notion. How much of practical use has theory generated of late? In what ways has our ability to manipulate the real world improved? It seems … not much. This is only a probable judgement, but enough to ask the question.

The above-quoted paper makes me wary. Sometimes application of theory has extended our understanding of the real world - when the map is accurate. When the map is less accurate than believed, calculation can take us down a blind alley.

We must always keep in mind - the map is not the territory!

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