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
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.
The translation of ‘near perfect’ works of literature almost inevitably always means loss of ‘literary value’ to the translated version, because a) of the translator’s inferior genius to the original author’s; and b) because that a work of great literature in its native language is like a beautiful large translucent gemstone set perfectly in a gorgeous ground; a gemstone which a translator has wrenched out of its surroundings, perhaps scratching or abrading it in the process, and s/he has then attempted to mount it in a recreated lookalike but inferior ground for it and under her/his own terms. In the same way, by the same arguments, any ‘translation’ of a theoretical model or blueprint from science of any kind into actual machinery or technology or hard substantial fact; is also inevitably liable always to entail a generation of some qualitative ‘loss’ arising; loss integral within the practical machinery created.
This might be a general rule: that drawing-board schemes when put into practice suffer loss of theoretical effectiveness of a course.
This is the case even when wear and tear, friction and other known entropies are all allowed for in the blueprints. It is not due to human error or to human imprecisions either. It is due to the inescapable fact of inherent imperfection being present in all sensorily perceived material things. And no matter how much one ‘allows’ for them, there will remain a heap of unknowns, perhaps unknowables, all of which carry their own levels and types of imperfection, all of which bears upon the effective practical usage of such material things. We just have to accept that we don’t know it all.
William Empson again:
“The waste remains, the waste remains and kills.”
This ‘waste’ which remains and ‘kills’ poses unanswerable questions; such as why items age; not in a mundane sense but in an enquiry which asks why ‘things’ are all ‘set up’ so as for them to decay and to deteriorate, in the way everything does on earth and in the known universe. One is able to see very clearly how the many and varied cultural myths of there having been once a ‘Golden Age’ are so prevalent and powerful.
Our machines and technological helps in their very shaping into their uses by their human manufacturers and designers may appear to buck this rule of ubiquitous deterioration; not because they don’t wear out; they do; but because they are systems and processes which harness physical and other science to produce a targeted and useful result for humankind. A washing machine; a telegraph pole. In their capability to harness the data of science for the furtherance of human convenience, these technological objects and their systems and processes might be seen as though they bore a simple balanced equation which says; as much as one harnesses processes and systems for human benefit; so an equivalent amount of deterioration and decay is unleashed in the unharnessed and unharnessable outputs, the ‘wastes’, from such devices and going into the ecosystem and into the order or the lack of order in the universe overall.
Living things are perhaps less ‘wasteful’ in this way in general; and partake in cycles of regeneration and decay; yet there remains a great deal of attrition amongst populations and species in order that the recycling systems of living things are able to be upheld. This attrition appears to be the price to be paid for the sake of preservation and maintenance of the natural recycling systems of living forms. And yet living things are yet caught up in, because they subsist physically in, the decay and deterioration of the non-living world and universe. These decay and deterioration are assisted greatly by the human-made technological unharnessed attritions arising out of and emanating from their machines and their gadgets as outputs.
My reasoning here may look like it is depending on some kind of inverse law to the scientific law of the conservation of energy; which is a law which is a generalisation and which has not nor can be tested to proof; only to its denial. This inverse law might say something like: One conserves energy always but this same conserved energy as we observe it over the longer term finds itself always more greatlydissipated than before its alteration or shift. This may not be the case of things. Things perhaps might be created or even be being created anew and possessing also tightly concentrated integrated parcels of energy which drive them, somewhere somehow in the realms of space or elsewhere. I don’t know.
But just as Newton’s Laws are useful and workable within our solar system and at our levels of human maginiiude; so to on the earth and in its vicinity, there is seen this ever and continuous winding down of material things going on ubiquitously all around. The great Shakespeare comments on his own observations saying:
“The cloud-capped towers, the gorgeous palaces,
The solemn temples, the great globe itself—
Yea, all which it inherit—shall dissolve,
And like this insubstantial pageant faded,
Leave not a rack behind. ”
Whereas mathematics, being ideal – by which I mean – in the mind – and so subsists as a series of deliberately discrete and segregated steps by which to try to harness the amorphous and continuous physical world; this means that maths indeed mat perhaps be eternal and unchanging in so far that it is ideal – but for these same reasons, and in the final instance maths remains useless to us as physical persons ‘crawling between earth and heaven’. This is because its format, its architecture, its foundation, just do not align with, do not fit, are incompatible with, the continuum and processes which are our material being and environment.
Take the figure 1. It is unity. It is static, unchanging, It cannot for these reasons ‘contain’ anything which we find in existence in the material world; since this material world is always ‘on the move’ messy and diffused; and perhaps as a set of pretty-fully interfused relationships which subsist interdependently in one another. Not like 1 at all.
But yet maybe there is hope in all this for we people? These imbalances and imperfections, and their messiness in their goings-on in the physical world seem to me to be necessary to what we might call ‘the learning curve of life’ for human beings. What I mean is that they are present not by logical or by physical necessity; although they impose on us logical and physical necessities. They are present as they are, but I do believe things have or had or have had a possibility of being/having been different in their set-up.
I believe that the set-up we have was chosen and has a purpose. I believe that that purpose is a pedagogical one; it is that these material imperfections and all the surprises and unpredictabilites, the knocks off course, and the inabilities to handle, that they bring to humankind one day after the next, are our life-teachers. They are our life-teachers knitted tightly into the fabric of material being so as to be there to ‘knock us into shape’ and so to make a people fit for living and for humanity.
They are then the instruments of a God of Love who has placed them here, so as to educate us in his ways and about his person. The static and non-dynamic differentiations of maths are a double sided blade; cutting one way to show us that perfect at least theoretically, is possible; and cutting the opposite way to show that human self-belief can be seductive and so draw a person away from what Wordsworth has called; ‘the essential passions of the heart’ and so they are able to puff us up in a false estimate of ourselves and of our capabilities. It is such a pride which in a weak sense is ‘necessary’ so that that ‘fall’ which is inherent within it and which ever follows on from it, is able to be efficacious for us and so to ‘awaken’ us a little more from our as yet unjettisoned pipe-dreams still cherished, and which we constructed for ourselves and for our own esteems since our beguiled childhoods.
The Book of Proverbs tells us that ‘To spare the rod is to spoil the child’. Our lives then, in the ways I have tried to show, inevitably experience God’s rod upon us, his children, by way of our being creatures caught in a pincer-grip between brute animal being and our hopes and spiritual dreams of comfort and beauty; dreams, hopes, which when they are given just a bit of thought, tend to conform what is possibly, what is even probably, the case – thought confirms the presence of a realm of fulfilled dreams and realised hopes, in the real and actually subsisting, being present in the purlieus of The Lord God.