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### Mathematics

 Mathematics is usually lumped in with sciences during our education but does that mean it is a science? The answer is no but the relationship that mathematics has with science, our everyday lives, and the universe as a whole is less than obvious.   Most people would say that mathematics is a tool that we use in science. It is the basis of our reasoning and logic. It is also the only subject where anything can be proved in the absolute sense. By contrast, in science, where nature or the universe dictates what is correct, our best theories and calculations are only ‘true’ until some future observation or measurement disproves them, and we can never be sure if and when that might happen.   So should we expect aliens from another world to share the same mathematical concepts as ourselves? Obviously their terminology and symbolism would be different but what about concepts such as arithmetic, geometry, calculus, set theory, logic, topology, symmetry, etc? It’s fair to assume that they would (or could) share them but does that mean there’s some fundamental significance to those concepts as opposed to them simply being inventions?   David Hilbert remarked that there’s nothing arbitrary about mathematics. In fact, mathematics is a hierarchy of tools – each level built on the principles and foundations of others. Between 1910 and 1913, Alfred North Whitehead and Bertrand Russell published a three-volume work on the foundations of mathematics called The Principia Mathematica. It was an attempt to derive all mathematical truths from a well-defined set of axioms and inference rules in symbolic logic.   OK, but there may not be one unique hierarchy that can be constructed from those basic principles. Maybe we’ve constructed it purposely to model the physical aspects of the universe around us. It’s sometimes hard to separate those physical aspects from the mathematical tools. A common faux pas in school science lessons goes something like ‘momentum is a vector’. Although the intention is fairly clear, the statement is not strictly true. Momentum is a physical quantity but a vector is a mathematical concept. A Vector can be used to model momentum under most circumstances but it isn’t the only mathematical tool used for that purpose. Momentum might be modelled using tensors, matrices, and even operators.   Consider a simple numeric scale for a moment. You may think this is so obvious that it doesn’t deserve any analysis.  Numbers are basic to our everyday lives and are essential for all things countable or measurable. The ‘natural numbers’ (1, 2, 3, etc) are the ordinary whole numbers used for counting. Zero wasn’t always part of this scale, though, and it took time for the concept to form. Other integer scales such as even numbers, primes numbers, etc., can all be derived from the concept of natural numbers. Although you might visualise them as a linear scale, as with a graphical axis, ideally they should be visualised as amounts, as with counting.   If you were measuring a distance then the natural numbers are no longer enough. You need a continuous numeric scale with indefinite subdivisions, or ‘real numbers’.  Now we visualise this scale as a physical line, primarily because that’s what it was designed to model. That set of numbers includes both rational numbers (i.e. quotients of two integers) and irrational numbers.   If we were measuring forwards or backwards from some reference point then we need to make the numbers signed (positive or negative) in order to distinguish the physical direction. Note that the sign is a mathematical attribute whereas direction is a physical one. Again we visualise this scale as a physical line since that’s how we’re applying it. This is in contrast to a usage for measuring the distance from the centre of a sphere, which would necessarily be non-negative. Negative numbers are also useful with counting but slightly harder to visualise. It then depends on the context of their application which mental model we use, e.g. profit/loss in accounting, or credit/debit in banking.   At this point you may be saying ‘OK, so all numbers lie on a continuous, signed numeric scale and we simply pick a subset to suit our needs’. Well, no, that’s an oversimplification. Consider ‘complex numbers’. These have a real and imaginary component, both of which are signed, continuous scales. Hence, we visualise them as a plane and represent the two components of the numbers using two separate axes.   You should be able to see where I’m taking this now. Our real numbers were designed to model the space we see around us, and hence we visualise them using a similar mental model – even when each number has more than a single, signed component. There are generalisations of complex numbers with four components (quaternions) and eight components (octonions). However, what use would we have for a number system with three directions taking the place of the binary positive/negative concept? How would we visualise it?   It’s worth spending a paragraph talking about infinity. Mathematics has the concept but it is not really a number. You cannot see it on a finite scale and you can’t really do any arithmetic with it. So does it exist in the universe? Well, possibly not. Physicists have tried to imagine the universe as “finite and unbounded” – in other words, no brick walls but no infinities either. In one dimension this is like the circumference of a circle, and in two dimensions it is like the surface of a sphere. We can’t really visualise the notion in three or four dimensions but we can describe it mathematically. This gets over big problems like calculating the average density of matter in the universe – which we know is a finite number - since otherwise we’d be dividing infinity by infinity, which we can’t do. The problem of infinities famously came up in quantum field theory and was a sign that the theory, as it was then, was flawed.   In summary, I am suggesting that our mathematics is based around our perception of the universe, and is designed to model that perception. It then makes sense when we visualise the mathematical tools using physical concepts like lines, planes, etc. The exception to this maybe the concept of pure logic and the basic Boolean notions of true & false. As indicated above, logic underpins all mathematics, but where does it arise from? The true/false notions map directly onto is/is-not in the universe itself, and the ultimate ‘is’ notion is Existence itself. The section on Existence identifies it as a unitary concept because it just “is”. All things within it, therefore, are ‘is’ and all things that are not within it (i.e. do not exist) are ‘is-not’. Since the picture of time presented in these pages is both geometrical and non-dynamic then nothing is really a ‘maybe’, or ‘might be later’; everything is either an ‘is’ or ‘is-not’. Hence, Boolean notions, and logic itself, stem from the nature of existence.   One final question here: can we ever be content with a purely mathematical description of reality? This is a question that frequently comes up because as physics advances, and it describes the workings of more-and-more phenomena in an all-encompassing fashion, then its notions become more-and-more abstract; describable only through mathematics. A related question was a cause of much debate in the early days of the quantum theory since some physicists — notably Bohr — believed that you could only describe the universe in terms of its observables, whereas others — notably Einstein — believed that there was an underlying reality, and a physicality that could be understood. For instance, Einstein was troubled by a probabilistic description of finding a particle at some location rather than understanding what a particle is, or what it was doing. Modern string theory has been criticised for the cavalier way that it introduces non-observable notions, such as extra dimensions, in order to make a consistent mathematical theory. What these issues have in common is the relationship between a mathematical and a physical description of the universe.   My answer to the above question is both yes and no: if you abandon any fundamental notion of Cause and effect, as I have done in these pages, then the answer must be yes. However, if you cling to some element of causality then it must be no. The problem is that mathematics cannot describe causality as we perceive it. The equation describing the path of a cannon ball does not say that it was ejected from a cannon to finish on the ground, or vice versa; it merely describes the total path. Causality is a set of semantics that we — as conscious entities — attach to our equations, and which are not an inherent part of them.