God and Mathematics
‘And brought to you of all that you question Him of. And if you count the favors of Allah, never will you be able to number them. Surely, man does - cross limits - cover the truth’. Quran 14:34
‘He is, Who made the sun a fire and the moon a shine, and measured for her, stages that you might know the number of years and the reckoning. Allah did not create all that except with truth. He detailed the revelations for people who have knowledge’. Quran 10:5
Great advances in mathematics in the last few centuries has brought thinkers to face a contest between the seeming perfection by which mathematics is able to explain the details of phenomenon we encounter around us in nature and the strange paucity of strength in the foundational structure of mathematics. Does this paucity alarm us, such that in spite of our enormous leaps in conceptualization with the aid of math, we may just be scratching on the surface of wholesome knowledge? And does this paucity bring us back begging at the doors of God? This will be the quest of this essay.
For many it may or not be possible to assimilate the essence of the idea that knowledge exists in several levels of understanding and that if one is able to grasp or is able to enter another level by any means of transport, one has entered another world of possibilities. The belief that such variant possibilities of existence actually surround us, possibly at every line of sight, has perhaps been made possible most profoundly by the advances of mathematics in the last two centuries.
The conceptual freedom of breaking off from the usual 3D level of experience and knowing of higher dimensions and the way things might work in them has opened upon human conscience new worlds in its imagination. This imagination is fed by the power to express new ideas in the various mathematical fields and spaces, the dimensions of which could stretch to infinity – by creating sets of axioms and mapping functions from one field to another, one would discover complex designs and possible meanings hitherto not thought of by the conscious thought. Moreover these complex possibilities show us worlds that we have no physical evidence of around us. This magical ability of mathematics urges us to question whether mathematics is an entity of the real world or of the abstract world of our imagination, and urges us to reanalyze the very foundations upon which mathematics is grounded.
Mathematics, at heart, is based upon abstractions of our thought. It is the ability of the human thought to assign names or symbols to things. These names and symbols are essentially a creation of the mind and the world of our thought is their primal abode. Again the human mind mysteriously synchronizes these subjective images with the sounds of the vocal cords and the drawings of the hands, and like this communication begins – qualia feels connected to other qualia.
In our everyday workings, not only do we assign names to objects but we also account for spatial, temporal and functional relations between them, and quantify and qualify them, and all this makes our language. Language also describes our inner sensations, our emotions and our fancies. And if we deem language to be ‘any form of communication’, then all forms of expressions like art and music enter its realm. Between this immense sea of the language, mathematics seems to be a specific part of it that deals only with quantities – and also with qualities where they can be quantized in rough scales.
Mathematics also presents two uniquely human things, firstly that the human race possesses the ‘intuition’ of counting and secondly its ability to generalize a universal system of counting over all sets-of-similar-things. Another uniquely human ability that precedes this act is the conceptual instinct of grouping things into sets based on one or more similarity. So the Sun, the Moon, the apple and a roundish stone can all be comfortably place in the set of circles… so much for how comfortable the mind is in doing the things it does…
The Greek mathematicians did humanity the favor of going down from the 3D world of usual experience to the one and two dimensional worlds that helped us understand our world by splitting it into dissectible wholes. These would then be rebuilt to make the holistic, concrete, real world that we are intrinsically sure of by living it every day. But the para-normal quirks of mathematics, this extra-ordinary faculty of the human thought, had been diagnosed as early as when the hypotenuse of a right-triangle with unit sides had been expounded upon.
The abstract basis of mathematics had thus been a matter of debate among thinker from the beginning. For instance, think of the relevance of the number Pi (22/7) in the area and circumference of a circle, if the circumference of a circle is really a multiple of Pi, a number that never ends, then the length of the circumference is also a number that never ends, something that is not real or objective! This factor of un-realness that repeatedly occurs in math raises the thought that math may be more of a fiction than a concrete thing.
Is mathematics a world imagined by the mind, a world of symbols that can relate to structures in the external world – or are numbers essentially the count-ability of the physical world, a trait of matter discovered by the exploring eye – or is it a language, like a medium through which two entities interact?
If mathematics was wholly subjective it would mean that it being a tool of our thought would sooner or later become obsolete like all forms of knowledge do with the evolution of knowledge that keeps becoming into new fashions of thinking in the two-pronged individual-plus-social processes of learning. If it was wholly a trait of the objective world then perhaps it is already an illusion for the fact that the mind never reads the world for what it really is rather for all x’s it reads y’s – and the mind is not equipped to determine what proportion of the whole truth ‘y’ is. And if it is indeed a language, one might ask if the talk is happening one-way or two-way, and again is this type of communication not subjective? In fact, ‘Fictionalism’ claims that the math-language is a story-making that helps us talk about the world; inside any specific story-making math is relevant and outside that story math is irrelevant.
If mathematics is a language, then it has often happened that its vocabulary has held certain numbers or concepts that did not match the real world at the time they were thought of. For instance the complex number ‘square root of minus one’ has long been known to mathematicians but always thought of as an abstract number with no real application, until recently when it has been found to be an essential value in quantum mechanics. As mathematicians play more and more with numbers, axioms and equations they come up with new mathematical vocabulary that might prove meaningful at a later time. This strange phenomenon brings us to the idea of an auto-mechanical nature of the mind that is predisposed to the tools that can read the book of nature, often it comes upon tools within its contemplation that will be of use in chapters that may be read only decades or generations later.
The concepts of irrational numbers and infinities yet again compel us to think about the truth and foundations of mathematics. One bethinks whether mathematics is a language that can speak of realities such as not experienced by us in our space-time world? The concrete world around us is made of wholes and fractions but not of things that are infinitely regressing into smaller and smaller bits, nor do we know of anything that is infinite in number of units or in size or in age. The real world seems to comprise of atoms, electrons and photons which are discretely present with a lot of void between them, yet mere numbers lead us to think that if we split 2.8x10-15, the radius of an electron, into yet another half or take yet a billionth slice of that distance, would we find yet some phenomenon, hitherto unknown, working in that premise.
Surely the mind can think that it is not necessary for every half of every distance to be occupied by some phenomenon, phenomenon can exist discretely and at distance if they have to – but what is amazing is the thought’s ability to visualize and know such a possibility like infinity – which has not remotely been physically perceived as yet. It seems that the thought just loves infinity, it yearns for infinity everywhere, in time, in space, in dimensions, in regress, in progress, in the infinitesimal. The human language that is able to express this love affair in words, has itself seemingly infinite possibilities of expression. One might conclude that if the world of our thoughts is such a world of possibilities then is the material world merely a set of limits tending to the nearest wholes over infinities tending to limitlessness?
Another dilemma of human thought is the fact that it does not certify a knowledge as ‘certain’ until it can put its finger on the thing by limiting it one way or the other. Albert Einstein famously said:
‘As far as the laws of mathematics refer to reality, they are not certain, and as far as they are certain, they do not refer to reality.’
Perhaps this means that when we talk of reality in the language of numbers, we are having a symbolic definition of reality and not reality itself. Perhaps because at the root mathematics is a generalization and however precise we get in our data, still we are generalizing.
Mathematics is the shorthand for a set of functions that we are intrinsically comfortable to perform, like adding things or subtracting things and then adding multiples of same numbers or subtracting multiples of same numbers. These simple processes when shuffled around in different sets of axioms come out like different sets of patterns from a kaleidoscope. And these simple processes of the mind when applied to phenomenon of nature, yield upon us an unprecedented truth that the whole of nature and even the whole universe is built upon the same language as is embedded in our conscious thought, the language of numbers. When we talk in terms of triangles and circles so does the world around us, when we talk in terms of probabilities so does nature, when we talk in terms of the Hilbert Spaces so does the Universe. It seems that as with every epoch man focuses on furtherer horizons and chooses to put limits upon newer possibilities, there he finds new realities, new ways the same Universe acts, as if there are so many faces of the mirror of reality, as many as one can think – in the language of mathematics – and are there other such languages!
The fact that abstract phenomenon in our mind have the ability to reconstruct the material world around us with accuracy raises questions like – how was the thought capable of abstract thinking, how are those abstracts so effective in measuring/defining physical phenomenon, while abstract objects create formulations in an abstract reality how does the reality of signs on a paper relate so truly with the reality of the physical world?
Usually the first thing we learn in the grammar of any language is the definition of nouns and the difference between common and proper. That is to say, the cursive language operates in a framework where it makes for generalization in the beginning. In mathematics too, we label these generalizations with numbers and move ahead one level in generalization from ‘a cat’ to ‘any number of cats’ regardless of their presence around us, therefore with the math-language we have come to a sort of imagination world where we can talk of substantial events which are not necessarily happening in real time and space. But this imagination field is equipped with symbols that are true/real in the sense that they give us knowledge about our surrounding, to a far higher level of accuracy than the bare senses are capable of.
Once you jump into this mathematical world of imagination you find that it is filled with patterns that can go as far as your imagination would take them. And you realize that the physical world is made of a limited set of possibilities while much of the rest of potential possibilities are made of void sets, so what happens is that while the patterns do fit our real world but on the fringes hint us of the other worlds which seem to be unoccupied.
In mathematics we make an axiom; we add with the number some adjectives or verbs as predicates to complete our concepts in mathematical sentences. But once these adjectives have taken the form of symbols they lose their adjectivity and turn into mere symbols with no meanings and a mere counting. Indeed this axiomatic world does not tell us the whole reality but does tell us some things about it with precision, yet what is striking is the fact that such a method based upon such overwhelming generalization can actually be precise!
So much so that every now and then it helps us decode the complex ways nature works around us. The universe is made up on patterns that can be decoded in the form of laws - laws are ideas that can take a mathematical formulation. But how does it come about that man alone is equipped with such higher conscious which has this tool of intuitions plus mathematics that working in combination - can decode the universe - which on the other hand also actually works on sets of laws that are so effectively decoded in precisely the tool possessed by the human conscience – the mathematical probability of such unique concurrence defies the idea of chance.
Has nature created unknowingly at its apex a creature that is amusing itself and its creator with a slow process of discovery and self-identification? Is this a sensible proposal that after making all the laws and putting all things to work, nature finally created someone who is totally ignorant of the laws and will make a journey of discovery now anew – is man the first one to consciously know the world and the universe hitherto unaware of their own existence? That would certainly not be a progressive thought on nature’s part, why would nature that evolves in progression, make a backwards stitch such as that?
But further onwards from precise measurement and decoding laws, the axiomatic nature of sets makes the unthinkable thinkable and gives a logical bases for existence of extra physical worlds. In fact physicists and mathematicians tend to make-believe that if the mathematical framework of a kind of existence exists and some plausible idea, with which the thought is comfortable, can fit into it, the probability of its existence in some form in time-space also exists, in a way that it would interact with other forms in some way, however remotely.
String Theory is a perfect example of mathematics led into a world of infinities that can well be imagined now, but cannot be verified as physically existent. But the power of String Theory, assuming its reach at the infinite, has not helped in capturing the ultimate truth of things, rather it leads to infinite number of universes, an idea which surely does not reduce the question of Ultimate Reality to ‘nothing’ rather it brings us to the need of a more immense God.
Abstract generality of mathematics deems it practically useful only in explaining extremely isolated bits of reality, therefore mathematical theorems identify only very specific properties of matter, digging deeper into the generalities of mathematics may bring to knowing more aspects of reality but will create a bigger maze of uncertainties around it… like when mathematics gives us a set of patterns, earlier it was eluded to that these patterns have been picked by observing the outside world, but as these patterns were studied in depth they showed element that were not in any way inspired by the physical observable world, and later on we are finding that mathematics is a world of its own, boundless and inexhaustible in patterns. Of every pattern that we find, only a small limit of it may fit into physical phenomenon that can be experimented on, but a width of which reflects a world we are unaware of. This shows that mathematics is a tool unique to human conscience, capable of measuring reality at different levels, touching infinity at one end and limited measurables at the other. Duality in the nature of mathematics matches the duality of matter and the soul.
Infinity is a phenomenon that takes the thought from a simple comprehendible 1, 2, 3 to an extension of the same numbers that is beyond comprehension. It gives us a hint that somehow, somewhere the physically real meets the physically unreal. How can the empirical lead to the sublime, the spiritual? Perhaps for the reason that in this whole complex maze of the Universe, there is one mirror of a conscience, where alone is found a ground so fertile as to experience within itself the duality of physical and abstract, of countable and infinite, of worldliness and Godliness.
So within this vast expense of mere matter, in the unique sprouting of life in a uniquely evolved planet and the apex of life in the form of human conscience and therein the unique experience of intuitive conception of abstract patterns that match with the underlying laws upon which all things move in set cycles of events in nature and in the Universe - prove to us that whatever led to the evolution of human kind and conscience, has made the Universe a book and conscience a reader - both conversing a like language, a language that is largely abstract but accommodates all material things as it has the ability to allocate symbols to things it experiences, inwardly or outwardly.
Does Math prove the existence of God? Is it so that in man’s infinitesimal existence within the Universe, there is a world of experience that encompasses all matter beyond spatial distances, binds within its experience phenomenon that clutch to the edges of the Universe. This dot of an existence, having the intuitive urge to find patterns, and in its urge to do so go as far in abstraction as String Theory and Hilbert Spaces – if theses phenomenon hold truth, that truth is nowhere in the physical but only in the abstract experience of the thinker, and if truth is something that we hold subjectively inside our experience then it is our abstract experience that has the greatest value of truth for us. And this urge for finding pattern in the Universe is also two-pronged, it is an urge to find ways to control on the one side and an urge to find the Ultimate Reality on the other. This urge that fuels the sublime tool of mathematics is our intrinsic experience, it is the framework of the thought, it is the way we think, the way we know the world, it is this inner framework which is built with the mortar of duality, the knowledge of matter and spirit, of visible and invisible – the visible is true, but in the invisible resides the more valuable truth.
Whatever is a design, even though invisible, like the Hilbert Space, must exist, at least as a possibility! Whatever does not follow any pattern or design must be a misconception – unless its pattern will be found later. Thought has the urge to find the pattern, which is actually a simplicity - complexity is only the state of not-knowing, once we understand the thing, it becomes simple. This is the law of the thought, expounding in any field of knowledge will eventually make it simpler and open newer horizons in it. This belief, this way of working of the thought never lets it rest or cease in its endeavors – it encounters itself with the most remote, the most challenging, because in its heart it knows that behind every wall of blinding there will be a simple truth – and it knows that the ultimate reality of all things will also turn out to be a simple and beautiful truth – and it will only stop there – at the Ultimate Reality!
‘He creates what He wills, and Allah is upon all things capable.’ Quran (5:17)