Monday 28 January 2019

Is Mathematics Invented or Discovered?

Posted by Keith Tidman

I’m a Platonist. Well, at least insofar as how mathematics is presumed ‘discovered’ and, in its being so, serves as the basis of reality. Mathematics, as the mother tongue of the sciences, is about how, on one important epistemological level, humankind seeks to understand the universe. To put this into context, the American physicist Eugene Wigner published a paper in 1960 whose title even referred to the ‘unreasonable effectiveness’ of mathematics, before trying to explain why it might be so. His English contemporary, Paul Dirac, dared to go a step farther, declaring, in a phrase with a theological and celestial ring, that ‘God used beautiful mathematics in creating the world’. All of which leads us to this consequential question: Is mathematics invented or discovered, and does mathematics underpin universal reality?
‘In every department of physical science, there is only so much science … as there is mathematics’ — Immanuel Kant
If mathematics is simply a tool of humanity that happens to align with and helps to describe the natural laws and organisation of the universe, then one might say that mathematics is invented. As such, math is an abstraction that reduces to mental constructs, expressed through globally agreed-upon symbols. In this capacity, these constructs serve — in the complex realm of human cognition and imagination — as a convenient expression of our reasoning and logic, to better grasp the natural world. According to this ‘anti-realist’ school of thought, it is through our probing that we observe the universe and that we then build mathematical formulae in order to describe what we see. Isaac Newton, for example, developed calculus to explain such things as the acceleration of objects and planetary orbits. Mathematicians sometimes refine their formulae later, to increasingly conform to what scientists learn about the universe over time. Another way to put it is that anti-realist theory is saying that without humankind around, mathematics would not exist, either. Yet, the flaw in this paradigm is that it leaves the foundation of reality unstated. It doesn’t meet Galileo’s incisive and ponderable observation that:
‘The book of nature is written in the language of mathematics.’
If, however, mathematics is regarded as the unshakably fundamental basis of the universe — whereby it acts as the native language of everything (embodying universal truths) — then humanity’s role becomes to discover the underlying numbers, equations, and axioms. According to this view, mathematics is intrinsic to nature and provides the building blocks — both proximate and ultimate — of the entire universe. An example consists of that part of the mathematics of Einstein’s theory of general relativity predicting the existence of ‘gravitational waves’; the presence of these waves would not be proven empirically until this century, through advanced technology and techniques. Per this ‘Platonic’ school of thought, the numbers and relationships associated with mathematics would nonetheless still exist, describing phenomena and governing how they interrelate, bringing a semblance of order to the universe — a math-based universe that would exist even absent humankind. After all, this underlying mathematics existed before humans arrived upon the scene — awaiting our discovery — and this mathematics will persist long after us.

If this Platonic theory is the correct way to look at reality, as I believe it is, then it’s worth taking the issue to the next level: the unique role of mathematics in formulating truth and serving as the underlying reality of the universe — both quantitative and qualitative. As Aristotle summed it up, the ‘principles of mathematics are the principles of all things’. Aristotle’s broad stroke foreshadowed the possibility of what millennia later became known in the mathematical and science world as a ‘theory of everything’, unifying all forces, including the still-defiant unification of quantum mechanics and relativity. 

As the Swedish-American cosmologist Max Tegmark provocatively put it, ‘There is only mathematics; that is all that exists’ — an unmistakably monist perspective. He colorfully goes on:
‘We all live in a gigantic mathematical object — one that’s more elaborate than a dodecahedron, and probably also more complex than objects with intimidating names such as Calabi-Yau manifolds, tensor bundles and Hilbert spaces, which appear in today’s most advanced physics theories. Everything in our world is purely mathematical— including you.’
The point is that mathematics doesn’t just provide ‘models’ of physical, qualitative, and relational reality; as Descartes suspected centuries ago, mathematics is reality.

Mathematics thus doesn’t care, if you will, what one might ‘believe’; it dispassionately performs its substratum role, regardless. The more we discover the universe’s mathematical basis, the more we build on an increasingly robust, accurate understanding of universal truths, and get ever nearer to an uncannily precise, clear window onto all reality — foundational to the universe. 

In this role, mathematics has enormous predictive capabilities that pave the way to its inexhaustibly revealing reality. An example is the mathematical hypothesis stating that a particular fundamental particle exists whose field is responsible for the existence of mass. The particle was theoretically predicted, in mathematical form, in the 1960s by British physicist Peter Higgs. Existence of the particle — named the Higgs boson — was confirmed by tests some fifty-plus years later. Likewise, Fermat’s famous last theorem, conjectured in 1637, was not proven mathematically until some 360 years later, in 1994 — yet the ‘truth value’ of the theorem nonetheless existed all along.

Underlying this discussion is the unsurprising observation by the early-20th-century philosopher Edmund Husserl, who noted, in understated fashion, that ‘Experience by itself is not science’ — while elsewhere his referring to ‘the profusion of insights’ that could be obtained from mathematical research. That process is one of discovery. Discovery, that is, of things that are true, even if we had not hitherto known them to be so. The ‘profusion of insights’ obtained in that mathematical manner renders a method that is complete and consistent enough to direct us to a category of understanding whereby all reality is mathematical reality.


Thomas O. Scarborough said...

Thank you, Keith. I consider that mathematics is a level of abstraction, which strips away (almost*) all sensible qualities from things. But there are two major problems here:

1. 'Stripping away' means that, in mathematics, we bracket out most of what is real, so creating a dangerous world. According to theoretical physicist Max Tegmark, we are losing the 'wisdom race' against the growing power of technology.

2. The units of mathematics may be anything we please: clouds with noses, ants that fall of the wall, and so on. Albert Einstein noted that a unit 'singles out a complex from nature'. We give too little attention to what our units are.

*For example, natural numbers are more likely to be used for counting things, while rational numbers are more likely to be used in construction.

Keith said...

‘I consider that mathematics is a level of abstraction, which strips away (almost) all sensible qualities from things’. I particularly like this sentence, Thomas, as I believe it’s central. If, as I do, one argues in favor of mathematics’ Platonic capacity as the notional footing of reality (all the while ‘discovered’ vice ‘invented’), mathematics is indeed reductionist. To your point, therefore, if we think about the etymology of the word ‘sensible’ in this sentence — which, if I’m not oversimplifying, might be equated to human experience — then as far as how we ‘sensibly’ experience reality, mathematics to that extent may fairly be construed as insufficient. The explanation of human experience is, of course, more layered than that.

In other words, it’s as if, for purposes of experience to happen, there’s a real-time translational mechanism — a smoothly continuously active function of human consciousness — that enables us to take what mathematical reality hands us and puts that reality into a form perceptible to us. The latter serves as a big part of what, aside from what’s known about reality’s mathematical ground-level underpinning — and what my essay argues is ‘discovered’, not ‘invented’ — forms our everyday, sense-based perception and understanding of that mathematical reality. That’s as true, I believe, for the mathematicians who are in the business of discovering the ‘mathematics of reality’ as it is for the rest of us.

Martin Cohen said...

Actually, Thomas' remark that " 'Stripping away' means that, in mathematics, we bracket out most of what is real" sounded very UN-philosophical to me! I don't mean that unkindly, let me explain. As Thomas will know, ever since the Pythagoreans and certainly since Plato, philosophy has talked airly about stripping things down and finding underlying reality, and by this they meant precisely mathematical reality. But to say that reality is here in the messy mix of experience is not entirely obvious either. As Plato says, the things we experience are transient and ambiguous. The abstractions of mathematics are eternal. Surely the latter entitites have a higher status that the former/

Keith said...

‘As Plato says, the things we experience are transient and ambiguous. The abstractions of mathematics are eternal. Surely the latter entities have a higher status than the former’. Yes, I agree, as I hope my essay conveyed, even if imperfectly: mathematics (with all its abstractions) is not only eternal; it’s objective and quintessentially foundational to reality. Once one reaches the mathematical underpinnings of reality, there is no further ‘stripping down’ of reality that’s possible. (No ‘bracketing out’, if that term is preferred.) Put another way, in my view there is no higher-order — or no more-foundational — aspect to reality than mathematics. Human experience of that reality, filtered and translated through the unavoidably distorting lens of one’s consciousness, leads to a subjective, imperfect, and thus lower-order reality — including, surely, the transience and ambiguity you refer to on Plato’s behalf. Yet somehow, despite their relative status, the reality of mathematics and the reality of ordinary human experience must, and in fact do, coexist.

Thomas O. Scarborough said...

We have opposing positions I see. Take the concept 'woman':

• The full concept: a living being that breathes, sleeps, looks at stars, operates devices, and so on to infinity.
• The semantic concept: adult human female.
• The mathematical concept: 1
And supposing that we kill her: 0.

And so we reduce reality to mathematical crudity. This is what has caused the havoc in our world, yet we pride ourselves in it, calling mathematics 'beautiful, 'eternal', and so on. It seems hard to deny the disaster of mathematics, when we look at the world today.

Martin Cohen said...

Erm... would a better example be something like a conch shell or a fern leaf? Each has a certain shape, which is indeed wonderful to look at.. and yet mathematics can add a new description which is elegantly simple.

Keith said...

Yours is an interesting perspective, Thomas. But I suppose, at the nub, I don’t see the mathematics–perception dynamic in such a binary way. I propose that the mathematical foundation of the multifarious reality of the woman — her enduring significance, both in life and in death — is itself more nuanced than the reductionist all-or-nothing toggling between ‘on’ (1) and ‘off’ (0), seemingly without accounting for the complex, probabilistic consequences to reality of the woman having lived.

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