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existed in this mass but in a confused and indistinguishable form there was an infinite number of homogeneous parts ὁμοιομερῆ as well as heterogeneous ones wikipedia his cosmos however was closed and finite the atomists 5 th century bce believed in an infinite void within which an infinite number of atoms swirled around all matter consisting of different combinations of those atoms aristotle 348 322 bce the most influential of greek philosophers down the millennia only accepted a potential infinity whilst firmly rejecting the idea of a real infinity it is always possible to think of a larger number for the number of times a magnitude can be bisected is infinite hence the infinite is potential never actual the number of parts that can be taken always surpasses any assigned number physics 207b8 aristotle thought that the acceptance of a real infinity would invariably lead to paradoxes and to illustrate this he recounts the paradoxes of zeno of elea c 495 c 430 bce which zeno elucidated in support of the monism of parmenides fl before 500 bce which described the world as one which could not change in anyway zeni s paradoxes are intended as proof that change is impossible the most famous of zeno s paradoxes is achilles and the tortoise which aristotle describes thus in a race the quickest runner can never over take the slowest since the pursuer must first reach the point whence the pursued started so that the slower must always hold a lead physics vi 9 239b15 a fuller more modern description wikipedia achilles races a tortoise giving the latter a head start step 1 achilles runs to the tortoise s starting point while the tortoise walks forward step 2 achilles advances to where the tortoise was at the end of step 1 while the tortoise goes yet farther step 3 achilles advances to where the tortoise was at the end of step 2 while the tortoise goes yet farther step 4 achilles advances to where the tortoise was at the end of step 3 while the tortoise goes yet farther and this continues infinitely basically step n 1 achilles advances to where the tortoise was at the end of step n while the tortoise goes yet farther this is of course the limit problem that caused so much trouble in the evolution of calculus which we will arrive at in a much later episode aristotle describes four more paradoxes involving infinity that he attributes to zeno in the elements euclid fl 300 bce also avoids real infinity presenting only potential infinity people often refer to his famous proof that there are infinite prime numbers but he doesn t prove that rather euclid in a wonderful reductio ad absurdum proof merely shows that cannot be a greatest a greatest prime number we assume there is a greatest prime number p we then create n the product of all the prime numbers from 1 to p and add one to the sum n 1x2x3x5x7x11 xp 1 this number is not divisible by any prime number as doing so always leaves a remainder of 1 this means either n is itself a prime or it is divisible by a prime number greater than p contradicting our initial assumption the place in basic arithmetic were infinity raises its mostly unwanted head is in irrational numbers there are various myths and legends about how the pythagoreans discovered the existence of irrational numbers thus destroying their own mystical belief that the world consists of the natural numbers the first irrational number discovered by them was the square root of 2 a major disaster as it occurs naturally as the diagonal of a square of unit side length according to the pythagorean theorem squares are everyday objects an alternative version of the legend says that the square root of 5 was the first irrational number discovered by them as this occurs naturally in the pentagram said to have been a secret symbol of pythagoreans either way ancient greek mathematicians were not happy with irrational numbers which they called incommensurable in their geometry i e taking our example of the diagonal of a square one can t measure it with a given fixed length so if the side of the square is 1 cm the diagonal cannot be measured by any rational multiple of 1 cm lengths one can t measure the one against the other interesting in this context is that the babylonians and the indians both had very good approximations to the square root of 2 the babylonian clay tablet ybc 7289 c 1800 1600 bc gives an approximation of the square root of two in four sexagesimal figures 1 24 51 10 which is accurate to about six decimal digits and is the closest possible three place sexagesimal representation of the square root of two 1 24 60 51 60 2 10 60 3 305470 216000 1 41421296 babylonian clay tablet ybc 7289 with annotations source wikimedia commons another early approximation is given in ancient indian mathematical texts the sulbasutras c 800 200 bc as follows increase the length of the side by its third and this third by its own fourth less the thirty fourth part of that fourth that is 1 1 3 1 3 4 1 3x4x34 577 408 1 4142156862745098039 an irrational number that caused a lot of problems in antiquity is the ratio of the diameter of a circle to its circumference which is incommensurable a number we now call pi or π how the greeks came to deal with incommensurable numbers and to produce an acceptable approximation for pi using a proto version of integration is the subject of the next episode in this series 1 kim plofker mathematics in india princeton university press 2009 pp 58 59 24 comments filed under history of mathematics july 2 2026 8 35 am just some guy with a blog when somebody wants to dispute something that i say about the history of science or the history of mathematics on social media and can t actually refute it with facts they often resort to a comment like the one that is the title of this post it is of course factually correct i am just some guy with a blog and i m even prepared to admit that the only real formal qualifications that i have are a very ancient set of very ropey a levels i m quite happy to admit that i am to a large extent an autodidact but i do have quite a large amount of formal academic training acquired over the years a lot of what i am now going to relate is already known to regular readers of this blog but i think it is worth repeating for any new readers who have joined since i last exposed the details of my somewhat unorthodox life journey i read my first book on the history of mathematics at about the age of five lancelot hogben s man must measure which i assume had belonged to my brother who was six years older than me i was a very precocious child and skipped not one but two grades in primary school and was still the most intelligent child in the class i spent three years in the highest grade and at the end was teaching myself from the teacher s books whilst the teacher taught the rest of the class there was however a small problem i couldn t spell with eleven i couldn t even spell my own surname and the proverbial drunken spider that had fallen into an ink well was a calligraphy master in comparison to me much value was attributed to spelling and good handwriting in my primary school and so i suffered with your intelligence i of course suffered from a learning disability two in fact but i didn t know that at the time and in fact only discovered it when i was already over fifty years old but more of that later by now i was avidly reading of all the articles on numbers number systems and mathematics in the children s brittanica a copy of which we had at home but despite having sailed through my eleven plus and an emergency appendectomy all within twelve hours i basically gave up on the school system because of my writing difficulties as i have related in an earlier post my mother died when i was fifteen and i ended up spending my fifth form and first year sixth as a boarder in the grammar school where i had previously been a day boy at the end of my fifth year my boarding house master asked me if i would be returning next year i replied that depends on the o level examiners he replied that in that case he didn t expect to see me he was more than somewhat surprised when i got by far and away the best o level results of any of the fifth form boarders i now entered the first year sixth and was confronted with a new school headmaster who played the role of the new broom that sweeps clean i shambling along in my usual couldn t care less style soon landed in his study he asked me what i wished to study at university and i replied history i had been deeply in love with history since i taught myself to read at three he naturally then asked why i was studying science a levels to which i replied because that s what i m good at he then suggested that i could study archaeology with science a levels i wasn t particularly keen on the idea my father was an archaeologist but i ended up going on my first excavation in the easter holidays i really enjoyed it and went on a much bigger and longer excavation in the summer holidays the university of cardiff excavation of the roman fort at usk in south wales i had in the meantime been expelled from my boarding school i was now living in london before being expelled during the school holidays and after full time sometime around then my father gave me his copy of eric temple bell s men of mathematics a very inspiring book but historically highly inaccurate i spent many years correcting in my head bell s historical errors however at the time i simply became addicted to the history of mathematics lock stock and barrel over the following years i would seek out and acquire books on the history of mathematics and later the history of science as well out of those early autodidact years i still have my copies of carl b boyer s a history of mathematics morris kline s mathematical thought from ancient to modern times arthur koestler s the sleepwalkers and jacob bronowski s the ascent of man back to the chronological narrative i spent my a level year at the then notorious holland park comprehensive the flagship of the labour government s comprehensive education policy i spent that school year smoking vast quantities of cannabis going to amazing rock concerts in the nights and dropping acid about once a week come on it was the schoolyear 69 70 and holland park ladbroke grove was the centre of the british counterculture in the 1960s and on into the 1970s it is not a surprise that my a level results were to put it mildly bad it s a fucking miracle that i got any a levels at all i still managed to get a place to study archaeology at university college cardiff in those days one of the leading universities for the subject i had spent another summer digging at usk before the a level results came out and a lecturer recommended my selection for a place based on having got to know me on those excavation i think it was more likely that he wanted to make an impression on my father who was much further up the greasy pole than he was so indirect nepotism i studied archaeology metallurgy and mathematics i quite enjoyed the metallurgy but found the mathematics which in that first year didn t go beyond what i had already learnt at school boring i was stimulated by the archaeology but had immense problems with the essays because of my writing problems however all in all i decided it was not where i wanted to be and dropped out at the end of the year over the next decade i continued to teach myself the histories of mathematics and science in 1976 i lived for a time in malmø in southern sweden and in the well stocked public library discovered both stephen körner s the philosophy of mathematics and karl popper s conjectures and refutations as a result of which i added the philosophies of mathematics and science to my private curriculum shortly after i returned to the uk an assistant in a small book shop who knew my somewhat specialised taste in reading matter drew my attention to imre lakatos proofs and refutations which as i ve documented elsewhere was possibly the most influential book in my life throughout the 1970s i did numerous things to pay the rent and buy groceries for several years i spent the summers working as a field archaeologist and at one point for a whole year working on archaeology sites you learn an awful lot about the mechanisms of history i also spent a lot of time time working as a sound and light technician in theatre and as a set designer and constructer i worked concerts as a stagehand and a ticket controller i did a stint as an industrial painter in the docks if somebody offers you work cleaning out the diesel tanks in the keel of a ship so that welding repairs can be carried out politely decline and walk away fast i did the obligatory six months working in cardiff s nineteenth century east moors steel works i worked in the quality control for the ancient coke ovens taking coal samples before they came into the ovens the top of a nineteenth century coke ovens at three o clock on a summer s night is like something out of dante s inferno belching sulphurous fumes and sheets of flame my final occupation before finally leaving cardiff was renovating second hand electric stoves for resale it says something about the economic state of the uk in the 70s that it was a roaring trade in 1980 i moved to germany when people ask why i answer quite honestly i went on holiday and never went back again i ve already documented the events leading up to this move so i won t repeat myself here i couldn t speak the language so i went to evening classes to learn it it was too slow for my taste and i asked if there was anything faster and got told there was a german as a foreign language course in erlangen at the local university i inquired and it turned out that i would have to register as a student to be able to do the german as a foreign language course i had always intended to return to university so i did the german high school system closes with a high school certificate das abitur which qualifies the holder to study anything at university my science a levels only counted as a restricted high school certificate so i registered to study mathematics after one year of the german as a foreign language course i took and passed the final exam and could begin my mathematics degree in those days the first degree at a german university was a master s complete with thesis which was expected to take four to five years but many students took longer in mathematics it was called a diploma and i registered with philosophy as my subsidiary as already related elsewhere my philosophy professor was a historian of logic and mathematics and from the very beginning i followed an intensive history and philosophy of science course of studies with a set of excellent teachers the maths department had absolutely no interest in the history of the discipline so after two and a half years i changed from mathematics to a master s in philosophy in order to do so i required at least one humanities a level so i took german a level cou...
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