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 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.

*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
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.

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