The Tenet Of Mathematical Logic
I’ve been itching to write this post for a long time but have always been held back by the breadth of the subject. It’s not something I can start and finish along the seam of the tear, there are intricacies I must overstep without a second thought while assuming that the reader knows all about them. That is the same liberty I will assume now, and bear with me if anything seems far-fetched or plain baloney.
When I was 12 years old, my father gifted me with this wonderful disc of the Encarta Encyclopedia, and for about 7 years after, I was glued to it for hours on end each day. It had all these interactive applications that facilitated quick learning and a smooth transition between theoretical inculcation and practical implementation. Two of those applications were my favorites: one had a timer and an assortment of brachiosaurus bones strewn around, all of whom I had to piece together correctly, and the other application had to do with assembling a telescope to observe planets at different distances from the Earth. Both of them were addictive – at least, to me – and even if I can’t say I learnt a lot, I definitely did get a long way in understanding how things work.
That’s an important thing: the correct answer to the how of everything. As much as we see things being executed in the thousands around us every day, there’s a stark difference in process flow philosophies between macroscopic and microscopic environments. As a plate is tossed into the air, it wobbles at first but then assumes a steadying spin it maintains as it falls down. How does one determine the spatial configuration of the disc at any given point of time? As a black hole gradually expands, its density experiences a tremendous increase, pushing it to expand more. We know of no all-consuming black holes today, so when does this process of accelerated expansion stop? We all know bananas are curved. Empirical observation gives us so much that they are so because one end grows faster than the other, but why is it that imbalanced growth rates give rise to a curved structure?
The one thing common to all three questions is the space of thought we all know as mathematics. It’s not just what we’ve been taught with textbooks and calculators. To a graduating engineer, mathematics is a government of numbers ruled equivocally by logic. At the same time, on the other side of the world, the same government of numbers is at work to resolve another intricacy in philosophy: it is a set of tools invented by us and for us, and there is more to that statement then meets the eye.
Being governed by logical rules and restrictions, it only means that the set of tools are immutable. However, for all conceivable practical purposes, the following can be said: if you conceive a space where rules defined by you alone dictate the consequences of a stimulus, then you can explore this space for yourself if only you can present specific rules that indicate conformity to mathematical logic. Let me give you an example. Say you have an argument to make in philosophy about some distant planet where things work differently (as differently as to seem alien to us), but still seem sensible. This is a philosophical thought-space. In this compartment of the universe, if there are some phenomena that show adherence to the fundamental mathematical operations (such as addition, subtraction, etc.), then you can conclude that, by extension, even if the environs of the planet seem strange, they are still conformant to mathematical logic and, therefore, within the limits of our understanding.
Here’s another (and more grounded) example: if 3D geometry is yet to be discovered, and you’re the first person to have noticed its significance, it becomes obvious that you have to either show that 3D geometry is an extension of 2D geometry or that they are both different, leading to the formation of two mathematical thought-spaces. Instead of undertaking the tedious task of defining your own set of rules to explore the new three-dimensional space, all you have to do is show that such operations as addition and subtraction also hold 3D space as they do in 2D space, and therefore they are governed by one and the same logic.
If you don’t find this juridicious elasticity fascinating, I do: it is an unorthodoxly manifested behavior that plays truant even in its immediate derivatives. That it graciously extends its umbra of logic to cover even the most obscure of sciences is testimony to its inalienable nature of operation, an assiduous engineer quickly opening windows through which humankind can understand more of this vast universe.