I’ve been reading Maths so as to be able to help educate my son in the subject. The guy who wrote the textbook I am using says that there are ‘rational’ numbers and ‘irrational’ numbers. He says an ‘irrational’ number is one which when expressed as a decimal figure carries on (it is thought) to infinity its decimal points, and there are no repeated recurring sequences in the decimal places such as one get when one divides,say, 10 by 3.
He cites pi and root2 as examples of ‘irrational’ numbers.
The maths text author writes that ‘irrational’ numbers are ‘naturally occurring’ – and this phrase ‘naturally occuring’ is not explained or defined or even discusssed by him. My maths text is an elementarty one – well, it is below undergraduate level; so I suppose such conundrums are considered by its author problems to be left alone until a higher level is reached by a student?
I guess if one includes everything which exists as being ‘naturally occurring’ then ‘irrational’ numbers do occur naturally. By definition. But this hardly helps -since everything else in Creation occurs naturally also. We have gone no further forward. Whether the author considers numbers and maths to be inside or else beyond ‘naturally occurring’ things I don’t know – but I am assuming that he must consider at least some maths and numbers as being outside and beyond, otherwise he has no point to make about ‘naturally occurring’ ‘irrational’ numbers.
And so, let’s for the sake of clarity divide up ‘naturally occurring’ things and call all these type of things the things which we perceive through our senses in the phenomenological world. This means that when I draw a right-angle triangle, with two sides of a single unit long, and a longer side; then the hypotenuse, this longer side will be length = root2. This is proven by use of Pythagoras Theorem which says that ‘ the square on the hypotenuse is the sum of the squares of the other two sides’ for any right-angled triangle. Thus 1 x 1 = 1 and 1 x 1 = 1 and 1 + 1 = 2, thus the hypotenuse is root2.
But is it? I know of (hard) experience that nothing in the ‘naturally occurring’ world is ever found to be perfect; that everything in the ‘naturally occurring’ world is imperfect. My knowledge of this inherent imperfection in temporal sublunary things takes me into a realm of faith. It presents a broad and absolute inductive statement based on everything in that realm which I have come across. If my statement seems controversial or jaundiced to you who read this and you disagree with me on it, I guess you are still young.
To be less flippant I can draw on good authorities to bolster my case; the whole corpus of thinking men and women since words have been recorded as human history – from Homer and The Upanishads, to Stephen Hawking and Brian Cox – and many many many in between.
I do think were a world vote to be taken on the question there would be a landside in my favour.
By use of simple logic, in the case of any triangle I might draw on a page, we can show that the pencil is bound not to ride over the paper at the same speed and in the exact same line however hard I try to be steady. The graphite in the pencil will not be of consistent equal hardness, and so will run off onto the page more easily and less easily accordingly. There may be wrinkles in the paper itselfunnoticeable to the human eye and these wrinkles may guide off course the pencil trace as it runs. There’s probably a great number of other ‘interferences’ at work on my drawing any triangle and which, all of them, contribute to my finished product being far from perfect.
I remember at school being asked to cut up a paper circle into slivers, and to glue them on a piece of paper, one up , one down, until all the slivers were stuck. So as to form a pretty ragged rectangle out of the original circle. Then, as a measurement of pi we were asked to find the area of the rectangle, and to divide the diameter of the original paper circle into the rectangle’s area. Of course even the best with the scissors and glue only drew quite nigh to pi in their calculations. And this is a very good example of how and why I want to argue that root2 and pi are not by any means ‘naturally occurring’.
I have a maths savvy friend tells me that once a fraction gets to over four decimal places on it is down to a molecular level of magnitude for normal household objects This means that in measuring say a shoe length a person is wasting her time to go beyond say two decimal points accuracy. To go to four points is OCD or autism.
I tell you this about decimal points because it gives you some idea of the weird world in which maths lives; where numbers are able to go on into lowest magnitudes not yet plumbed by physical science; and perhaps maybe are redundant anyway for physical scientific purposes? It is to this absurd degree one has to go to get as near to ‘mathematical perfection’ as is humanly possible.
But in the rough and ready ‘real’ world there appears to me to be a great divorce from the world of maths, which seems to me to be pretty altogether detached from the sensory world. Of course people do successfully apply maths as engineers as architects, as motor manufacturers and so on to this sensory world and so make things which work – and all out of the use of a study of maths. They work, these appliances and gadgets and machines I believe simply because of the fact of four decimal places rendering measurements at a molecular level, because it looks to me that this fact leads to a conclusion that the margins for error and for tolerances are so large in the vast majority of practical applications of maths, that a use of maths in making things is quite ‘safe’.
Yet there seems to be no place at which the realms of sublunary common sensory presences meet with number and calculation. Not even tangentially. I do believe that items in sensory existence are a continuum, and a process; whereas maths is not this at all; maths is about dividing up such a continuum or process into finite discrete bits for use in their manipulation by human minds. It’s the logic of the tortoise and the hare to try to make the pint pot of maths contain the quart quantity of sensory phenomena.
Interestingly my friend whom I have consulted on these maths topics says that any isosceles right-angled triangle of legs or bases of any length will carry an hypotenuse whose length is a multiple of root2. This means then that the ‘irrational’ number condition is built into the complete species of right-angled isosceles triangles. There is no right-angled isosceles triangle which is not bearing an hypotenuse which is not a multiple of root2. (This is all done on an (imaginary) perfectly flat surface of course)
This plethora of extensions of decimal points to infinity which characterises ‘irrational’ numbers then is not found at all as ‘naturally occurring’; not even when we as people dissect and segregate into items phenomena present in the continuum and process of sublunary sense data, by us naming them, and considering them to be wholly discrete objects – ‘object this’ and ’object that’ etc. There may be a case for saying that this ability and need people have to divide up, for dividing up, and segregating into discrete labelled objects the sensory world, is innate and an imposition which we impose upon our perceptions so as to be able to order and store and recall and so manipulate our ‘experience’.
It may well be the case that the world of ‘naturally occurring phenomena’ in fact carries no objects per se apart from those which humans have designated to be such. The South American Indians who saw the Conquistadors as centaurs because they had never seen a man on a horse; the incongruities between different languages in regard to how they dissect and order the world; both these facts are evidence which tends to uphold the idea that there is merely a continuum in process rather than a bunch of whole discrete objects around us.
We are able to extrapolate this observation on maths and its dividing experience into a fragmentation into the whole of the natural sciences altogether. The conditions for scientific practical experimentation are considered to be,as it were, conducted in a ‘vacuum’, and so are regardless of a host of ‘things’ which (at least presently) are considered as being of no account, irrelevant to any experiment. The words of William Empson are pertinent here. In regard to our human need to select and to screen-out and to marshal and sift our thoughts and consciousness he wrote:
“We don’t want a madhouse and the whole thing here.”
The world, this created sensory existence, is too large to unwieldy for us to allow its influences in toto into our practical experiments and observations; no single man or woman is capable of such a thing; and maybe (because of this fact) no machine is either?
Perhaps too we as a scientific people and living in a scientific age, we have been seduced by our own apparent successes in science and its applications? We perhaps too much assume that that apparent perfection available to us in ideas and in the workings of the mind, in maths and in all science, is readily translatable into products and services and into Cern projects and Voyager projects without loss, or at least with a potential for being without loss in the transition.
A person working all day in the realms of thought and ideas is very likely to become prone to espousing beliefs that perfection is at his or her fingertips; and that s/he has only ‘to translate it into material things’ and the world’s problems largely will be solved.
To be continued...
Visit our metanomalies blog to read the whole article: https://metanomalies.com/root-2/
Nice post! Math is trully important for everything around us, science and everything. Math is life thought this vast universe all of us live in.
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