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shown that there exists at least one additional prime number not included in this list let p be the product of all the prime numbers in the list p p 1 p 2 p n let q p 1 since q is either prime or not if q is prime then there is at least one more prime that is not in the list namely q itself if q is not prime then some prime factor p divides q if this factor p were in our list then it would also divide p since p is the product of every number in the list if p divides p and q then p must also divide the difference 4 of the two numbers which is p 1 p or just 1 since no prime number divides 1 p cannot be in the list this means that at least one more prime number exists that is not in the list this proves that for every finite list of prime numbers there is a prime number not in the list 5 in the original work euclid denoted the arbitrary finite set of prime numbers as a b γ 6 euclid is often erroneously reported to have proved this result by cobtradiction beginning with the assumption that the finite set initially considered contains all prime numbers though it is actually a proof by cases a direct proof method the philosopher torkel frensén in a book on logic states euclid s proof that there are infinitely many primes is not an indirect proof the argument is sometimes formulated as an indirect proof by replacing it with the assumption suppose q 1 q n are all the primes however since this assumption isn t even used in the proof the reformulation is pointless so we actually turn to the elements of euclid in the translation by heath book ix proposition 20 1 what is most interesting about this very simple almost skeletal proof is that it doesn t directly use numbers but non quantified mathematical magnitudes straight line segments which were introduced by eudoxus of cnidus c 390 c 340 bce in order to work with geometrical entities thereby avoiding irrational numbers they are described in detail book v of euclid s elements i shall be discussing them in the next post in this series this detail in the elements is not mentioned by any of those insisting on returning to or maintaining euclid s original proof by contradiction rather than the later reductio ad absurdum proof so there you have a lot of detail about euclid s proof that there is no greatest prime number which of course leads to the conclusion that there are infinitely many prime numbers a conclusion that euclid studiously avoided 1 euclid the thirteen books of the elements vol 2 books iii ix translated with introduction and commentary by sir thomas l heath second edition unabridged dover 5 comments filed under history of mathematics july 9 2026 7 53 am take it to the limit one more time a history of calculus i the concept of infinity lies at the heart of calculus the infinitely large and the infinitely small also known as the infinitesimal in fact in the early days of calculus at the beginning of the eighteenth century it was known as infinitesimal calculus springing forward to my schooldays integral calculus was introduced by looking at the area under a curve produced by a function drawn in a cartesian coordinate system this area between two given values on the x coordinate in divided in to equal rectangles the area of under the curve between those coordinates is equal to the sum of the areas of the rectangles plus the areas of the little triangles left between the top of the rectangles and the curve which are not easy to calculate by making the rectangles stepwise narrower those triangles become smaller and smaller until the rectangles are infinitesimally wide when the triangles cease to exist and the area under the curve becomes the sum of the areas of the rectangles of course that final step taking the sum to the limit causes no end of philosophical problems how can you add areas that no longer exist as each infinitesimal rectangle is a single point of the curve and any line segment consists of an infinite number of points how are you supposed to find the sum the search for a solution to these philosophical problems will form part of the end of this series but for now we will return to the classroom and our calculus 101 course the derivative is often described as the instantaneous rate of change the ratio of the instantaneous change in the dependent variable to that of the independent variable wikipedia for a given point on the graph of a function the derivative is the slope of the tangent to the curve at that point finding it is done by drawing a chord to the curve from that point the slope of which can be found geometrically then the length of the cord is reduced to zero taking the limit and the slope is now equal to that of the desired tangent one way to think of the derivative is as the ratio of an infinitesimal change in the output of the function to an infinitesimal change in its input leaving our calculus course we now go back in time and look at how early mathematical cultures first encountered infinity and how they regarded it infinity raises its head at the very beginning of elementary arithmetic so the very beginning of human activity that might be termed mathematical arithmetic starts with the counting numbers one two three etc as soon as a culture develops a system for generating the counting number they encounter a potential infinity there is no greatest number one can always add one more to any number reached to produce a bigger one a never ending or infinite process going in the other direction if i have a given magnitude i can in theory produce a magnitude half as big and i can seemingly keep on halving the resulting magnitude without end creating a potentially infinitely small magnitude so how did those early mathematical cultures handle infinity when it comes into view there is no known treatment of the infinitely large or infinitely small in early babylonian mathematics that doesn t necessarily mean they didn t have one our knowledge of babylonian mathematics consists of what has survived on clay tablets and they are still literally tons of those clay tablets that have not yet been deciphered so maybe somewhere ancient chinese philosophy had the concept wuji which means limitless and in neo confusion cosmology came to mean the primordial universe which we don t need to investigate further in mathematics infinity appeared comparatively late and we will be looking at its appearance later the ancient egyptians had a personification of infinity or eternity heh his name originally meant flood referring to the watery chaos nu that the egyptians believed existed before the creation of the world the egyptians envisioned this chaos as infinite in contrast with the finite created world so heh personified this aspect of the primordial waters in mathematics the papyrus rhind contains a problem that requires the sum of an infinite series but carried out algorithmically without any real consideration on the concept of infinity as might be expected the ancient indians dealt extensively with the concept of infinity the vedic literature has the sanskrit term ananta meaning endless it is one of the many names of the god vishnu ananta is one of four types of objects or categories of being ananta has a beginning but no end nitya has neither beginning nor an end anitya has a beginning and an end anadi has no beginning but has an end according to jainism the pure soul of each life form is ananta gyana endless knowledge ananta darshana endless perception ananta caritra endless consciousness ananta sukha endless bliss wikipedia jain mathematics c 4 th 3 rd century bce recognised several different forms of infinity the numerable and the infinite mentioned in this passage were not the only recognised categories for the size of numbers in jainism numbers could be numerable innumerable i e finite but impossible to denote by a named quantity or infinite endless there were also different varieties of infinity described as infinity in one way infinite in two ways infinite in partial extent infinite in entire extent and eternally infinite special computational techniques were involved in dealing with these concepts 1 as we will see in a later episode there were significant developments in india mathematics connected to infinity in the medieval period we now turn to the ancient greeks and the infinite infinity as a philosophical concept appears amongst the pre socratics anaximander c 610 c 546 used the term apeiron meaning unbounded indefinite to designate the original principle perhaps something like the void in the christian creation myth but there is much discussion amongst the expert what exactly he said or meant anaxagoras 500 428 bce believed that the matter in the universe has an innate capacity for infinite division according to anaxagoras all things have existed in some way from the beginning but originally they existed in infinitesimally small fragments of themselves endless in number and inextricably combined throughout the universe all things 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 a...
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