Monday, 20 July 2015

Do Numbers Exist?

Around the world you’ll find a bunch of people – notably physicists, mathematicians and philosophers – who think mathematics is no more than just a human invention; and you’ll find a bunch of other people – notably physicists, mathematicians and philosophers – who have worked out precisely why mathematics isn’t just a human invention. The most probable reason why the members of the first group don’t realise they are wrong is because they have taken a partial description of mathematical reality, namely the descriptive representation of numbers we see in the form of lines and squiggles (like the number 1, 2, 3, 4 etc), and mistaken it for the whole of mathematical reality itself.

Mathematics is not a mere human construct – it is both a discovery and a construction, and the two interlock. For example, we discover mathematics when we draw triangles on a piece of paper (although triangles are an ideal shape, as are circles and straight lines - they don't really exist except as ideation), and yet we create it when we undertake geometrical elaborations that are a furtherance of that initial drawing. We discover mathematics further when we find out that nature realises these elaborated geometries when we observe the gravitational field, but then we see that nature places limitations on our geometries in that not all of them return accurate profiles of the gravitational field. The discovery and the construction are two wings of the same empirical bird - and both are mutually complementary.
The partial description of mathematical reality consists of the symbols we’ve constructed to represent numbers. So, for example, with the integers we’ve grabbed a pen and paper and constructed those lines and squiggles to make symbols that read 1,2,3,4, and so on, and we use them to do our sums. Naturally, those numbers in their symbolic form do not exist outside of the minds that construct them - they are descriptions of mathematical reality out there in the universe (and probably beyond). But they do exist out there insofar as what the symbols represent are actually facts about the universe, how physical things behave, and about the patterns, laws and regularities to which they conform from within the wider reality of the mathematical whole.
While they represent facts about physical reality, physical reality isn’t the whole story of mathematics, it is only a tiny fraction of it. Those that think mathematics is only a human construction have mistakenly focused only on that tiny fraction - our creation of those symbolic representations - while ignoring (or not realising) that what those symbolic representations represent is the most important part of their construction. To use an analogy, such people are like cartographers who design a map of a territory and then as soon as the map is finished they proceed to claim that the territory to which the map relates no longer exists. It's a common error to believe that mathematics represents a map and that the territory is the physical universe, but the reality is it's the other way round - the physical reality is the map and the mathematics is the territory.   

Once you see that the universe is a mathematical object it is no longer tenable to see it as merely a physical object. The reason we see it as a physical object in the first place is because we are physical objects. That it is a physical object is down to our human perceptions of the physical world – there is no physical picture of the universe as humans perceive it anywhere except in human minds. We humans live I what I call a mental matrix – being physical beings we are locked into a world in which the empirical, the physical, the metaphysical and the humanly constructed mathematical symbols are different aspects of the much broader mathematical reality that has an existence more primary than the mere physical.

That is why we see the world in terms of physics, chemistry, biology, solids, liquids and gases - it is because we are physical beings in a physical world locked in to physical perceptions, rather like a man who has never seen Texas (call him Bob) holding a map of Texas is locked-in to his two dimensional perception of it. We have no choice but to see the world that way, but this limitation causes people to mistake the whole mathematical reality as being just the physical reality we perceive. Getting this right can, and should, open up your minds to a more stupendous view of reality, which gets the analogy of the map, the navigation and the territory the right way around. Physics is to us as the map of Texas is to Bob, and mathematical reality is to us as Texas is to Bob.

I never cease to be amazed by people who uncritically accept the reality of physics but tell us that the universe is not a mathematical object because they prefer to believe that mathematical objects are only ideas about the universe. Physical things 'are' mathematical objects - so denying the reality of mathematical objects while accepting physical things is like denying the land of Texas exists while accepting that the map of Texas is accurate.

Someone once said to me, in opposition, that if mathematics is more complex than what we can construct we'd be unaware of that mathematics. I had to thank him for proving my point very well - we are unaware of it - that's the big clue. The upshot is, we've created symbols to represent part of what we know to be true of mathematical reality, but we keep discovering that the maps to the territory point to complexities far beyond what we can construct. That is because the particular universe that we live in is a mathematical object is only a tiny one, and vastly unrepresentative of mathematical reality as a whole, just as a map of Texas is vastly unrepresentative of Texas as a whole. Mathematics is the territory, and we construct the maps. Just like in cartography, the maps are only a sparse representation of the territory in actuality. The maps are constructed through our locked-in perception that is physical reality, and the symbols are part of that navigation. 

Furthermore, mathematics cannot be ‘just’ a human invention because it gives the appearance of being something with fixed laws that were always there to be discovered. That is to say, the stability of the natural numbers is more stable than the physical reality with which we interface. If we rewound back time a few billion years to just before the occurrence of abiogenesis on earth, 2 + 3 = 5 would still be true, even if there were no human minds to think up the maps to this truth.  A planet with 2 + 3 moons would still have 5 moons, with or without human senses*. 

There are people who will happily tell you that numbers don't exist in the same way that, say, trees and water exist. That’s true, but that doesn’t tell us much at all; the universe doesn't exist in the same way that trees and water exist, but we believe it exists nonetheless. Numbers don't exist in the same way that trees, water or the universe exists, but they do exist. 

It is a truly enlightening thing to realise that the universe is a mathematical object, and that physics is only a tiny representation of mathematics as a whole; and equally special to see oneself in that picture as mere cartographers trying to navigate the mathematical landscape with symbolic constructions we call 'physical things'.

* On this, watch out for when some mathematicians try to fool you by saying that 2 + 3 doesn't always equal 5.  When mathematicians come out with things like 2 + 3 isn't always 5, they mean there is more than one way to skin a cat.  They are basically saying there are multiple ways to express mathematical facts about numbers by changing the system.  For example, 3 is a prime number, but is it a prime number in every conceivable situation?  The answer is, it all depends on the number rings one chooses to use.  3 is a prime number because using one system it can only ever be divided by 1 or itself. But 3 can be maid into something else - say, 3 = ( 1 + i Sqrt[2]) (1 – i Sqrt[2])*.  * Sqrt[2] is the square root of 2 which equals 1.414. In doing that I'm changing the rules of the game to a complex number (I'm moving the goal posts).  But if we agree beforehand that we are only using non-imaginary integers, then 3 is always a prime number, and 2 + 3 always equal 5. Consider English grammar as an analogy – you are free to change the symbols, you can even change the rules if you want.  You can have full stops in the middle of a sentence; you can finish questions without a question mark, you can even replace punctuation marks with reiteration, but you’ll no longer be sticking to the rules of grammar that we all understand.  Similarly, with mathematics, you can start a new system of arithmetic where two apples plus two apples equals five apples, but you’ll be playing by different rules to the rest of us.  Sure, you can express numbers how you like (Boolean, modular, binary, etc), but you cannot change the fundamental rules of arithmetic.