robin2

Berlin vs. Göttingen

For me, the eye-opener in The Architecture of Modern Mathematics was the suggestion that the 20^th disputes of the foundations of maths (platonism vs. intuitionism, etc) were, in part, confusing after-echoes of a pivotal conflict in late 19^th German mathematics between the the work done in the universities of Berlin and Göttingen: more specifically, between the different approaches to complex analysis taken by Karl Weierstrass and Bernhard Riemann.

There is a story about the rigorization of the calculus wherein doubts over the status of its concepts where resolved by progressively expelling geometric intuitions in favour of strict arithmeticization: starting with Cauchy, but happening decisively with Weierstrass.

Riemann's work went the other way: finding new forms of geometric thinking with which to understand complex analysis. Weierstrass, for whom geometry didn't even properly belong to pure mathematics, would have none of this. So whilst Riemann's results were in advance of others, the party line in Berlin was that his methods were invalid, and that anything of genuine value that he might produce would have to be re-worked with appropriate rigour.

The conflict can be characterised as between computational and conceptual understandings of mathematics. So in Berlin, functions were taken as power series, and the treatment of potentially problematic mathematical entities (irrationals, imaginary numbers) was to say that if they could be represented canonically and if it was known how to calculate with them then that was all that mattered. Whereas in Göttingen great value was placed on having representation-independent understanding of mathematical entities, and on being able to obtain results by “pure thought” rather than by calculation.

Riemann died relatively young, and there were differences amongst those who regarded themselves as carrying on his work; one can distinguish an intuitive-conceptual (Klein) from an axiomatic-conceptual one (Hilbert). Also, it seems that part of the motivation for early 20^th century investigations of the foundations of mathematics was not so much carrying forward the Weierstrassian spirit as attempts by those of the Göttingen party to show that rigour wasn't confined to Berlin.

This three-fold distinction makes much more sense to me than intuitionism/formalism/platonism. (This latter trinity is just a bit odd, intuitionistic logic being no less formal that classical logic.) It might be interesting to see how it could be projected further back: e.g. John Wallis the computationalist, Leibniz the axiomatic-conceptualist and Principia-era Newton the intuitive-conceptualist.

Screwdrivers

Years ago I read a book that was the transcript of some seminar, chaired by Umberto Eco, that was about the relationship between readers' interpretations of books and authors' intentions. (There may have been more to it than that: I forget.) In it, Richard Rorty took the view that a reader could interpret a book any way they saw fit, and no-one could rightly say that they'd got it wrong.

Rorty gave the following analogy. A spreadsheet is a piece of software that is intended to be used to do calculations, whereas a word-processor is intended to be used for writing documents. However, if he found that he could write documents perfectly well using a spreadsheet then that would be a perfectly valid thing to do.

Someone countered, saying that although Rorty might not be interested in the difference between a spreadsheet and a word-processor, there was a legitimate field of human enquiry—Computer Science—that was concerned with establishing these distinctions. (So that is what computer scientists do.)

My thought was: “It won't sound so clever when you're explaining it to I.T. support. And I bet you'd use a screwdriver to a open paint can.”

More recently I was listening to a recording of Hubert Dreyfus' introductory lectures on Heidegger. In talking about the nature of equipment, there was (inevitably) a discussion of hammers: a tool is defined by its use, and so a hammer is at its hammer-most when its on the job—being used for hammering—but that this won't show up if it considered as an object with objective properties.

One student wanted to make a point about equipment being used for more than one thing. Presumably unfamiliar with claw-hammers, he gave the example of a screwdriver, which is normally used for driving screws but that someone might use as a lever, when taking the lid of a paint can.

There are a number of questions that might be raised at this point:

Does the discovery of new affordances in a piece of equipment support Graham Harman's view that an object is never exhausted by any definition that might be given for it, whether by use or by description?
Conversely, can't it be argued that the fact that, e.g., an object that is a hammer can be other things too doesn't affect what it is as a hammer?
Does that last response not risk replacing a piece of equipment with the idea that we have of it?

However, there is a more fundamental point to be made, that philosophers and their students might want to consider. DON'T OPEN PAINT CANS WITH A SCREWDRIVER. IT BREAKS THE BLADE, AND DECENT SCREWDRIVERS AREN'T CHEAP. USE A PAINT-STIRRER, A 2P COIN, THE HANDLE OF A FORK, OR ANYTHING ELSE. JUST NOT A SCREWDRIVER.

Thank you. I feel better now.

Probability and the infinite worlds

In Potentiality and virtuality (Collapse II), Quentin Meillassoux discusses Hume's Problem: there is no logical reason to believe in fixed laws of nature, as any formulation of such laws would be on the basis of empirical observation, and there there is no logical reason to think that the past is a good guide to the future. On this account, there is always the possibility that the ‘laws’ of nature might suddenly change tomorrow.

Meillassoux takes on a counter-argument that if changes in the laws of nature were possible then they surely would have happened. They haven't, so the possibility would have to be regarded as so unlikely as to be non-existent. This can be re-stated so as to make it's circularity obvious: the past can be taken to be a reliable guide to the future because it always has been previously.

Meillassoux takes a different tack. In order to be able to assess how likely it is that the laws of nature could change—given that no such changes have taken place so far—we would have to know the prior probability of the universe having fixed laws or not. (Similarly, if someone wanted to ascribe significance that physical constants are “just right” to allow for the development of life then they'd have to understand the probabilities involved there.)

So one has to consider all the possible universes to know how significant it is that we got this one. For Meillassoux, merely declaring that all this is outside the valid scope of probabilistic reasoning, or that the full range of possible universes is empirically unknowable, would be too weak a result. It would leave Hume's Problem where it was: as a lack of certain knowledge rather than as positive knowledge of a lack of certainty. So instead he contends that it would not be possible to draw conclusions from the universe being as we find it, even with a purely theoretical understanding of the range of possible universes.

At this point I lose the thread of the argument. For Meillassoux, if we claim to imagine an infinity of possible universes then we must acknowledge that—since Cantor—mathematics provides infinities in different sizes. Therefore “since there is no reason […] to choose one infinity rather than another” then probabilistic reasoning is impossible.

I'm obviously missing something fairly vital here. The article expands on the theme of sets and infinity, so this must be key to the point that is being made. But at this juncture the argument comes across like someone trying to crack a walnut with a sledgehammer, and missing. For now I'll just counter with the two points I've got at the moment.

The first isn't to do with the counter-argument to Hume's Problem, but with the Goldilocks argument alluded to above, that Meillassoux treats as being part of the same “probabilistic sophism”. The argument as I understand it is that if the constants of physical laws had been only slightly different from what they are then life, including human life, would have been impossible: and so it strains credulity that they could be as they are by chance. So what is being envisaged here is that the range of possible universes is determined by the ranges of possible values for a finite set of real-valued physical constants. In which case it isn't true that the infinity of possible universes cannot be known: it is that of the continuum.

The second is a bit whimsical. It seems strange to suggest that it might be theoretically possible to know everything about the possible universes apart from the size of infinity of possibilities, but suppose this is granted. Now suppose the possibilities can be ordered into a sequence, so that each possibility can be assigned an ordinal (which may be transfinite). Also, suppose the following:

Every universe corresponding to a limit ordinal (including 0) has fixed laws
Every universe that immediately follows one with fixed laws is has laws that change, but just once
Every universe that immediately follows one with laws that change once has laws that change all the time
Every universe that has laws that change all the time has fixed laws

Assuming that the number of possibilities isn't finite, and that they are all equally likely, then the probability of the universe having fixed laws would surely be 1 in 3, regardless of the exact length of the sequence.

Holiday Reading

This year my holiday reading included bits of Collapse covering speculative realism (thanks, Robin), some old papers by Rodney Brookes on Artificial Intelligence and robotics, and The architecture of modern mathematics—Essays in History and Philosophy eds. José Ferreriós and Jeremy Gray. All thought-provoking stuff. Unfortunately I've lost track of the thoughts provoked. I'll see what I can do, though.

Biology

Note: I wrote this a while back after having read the interview with Alain Badiou in Collapse I, but didn't post. I don't know quite what I mean by “Bergson, of all people.”

Badiou's claim that biology cannot yet be counted as a science shows that he has a rather idiosyncratic notion what science is. The complaint is that biology is a morass of observations and specific theories: these may give rise to plenty of useful knowledge, but nonetheless biology lacks an over-arching conceptualisation that would give the subject meaningful order. The concepts of biology “fail completely to present the phenomena concerned in the register of eternal truths.”

I don't want to get into the question of whether the requirement for eternal truths is a merely philosophical imposition; nor the question of whether Darwinism might speak in the register Badiou wants to hear (nor whether this would be a good or a bad thing). Instead I'll reminisce for a bit about why I never cared much for biology at school.

There were superficial reasons; such as the variety of dead things in pickling jars displayed on rickety shelving around the lab. But it was also that what we were taught seemed to be without rhyme or reason. Whereas with chemistry almost everything at that time seemed to be tied in with the periodicity of the elements, biology was just one thing after another: pollination, the set of little bones in the ear, the carbon cycle, the circulation of the blood, etc, etc. It sometimes seemed that the main skill required was to be able to pick up on all the vocabulary involved.

In reality the main skill was to be able to correctly label the diagrams that featured in the syllabus: this much as explained to us by one of our biology teachers. She also suggested that it was a good idea to remember the number of labels that each of the set diagrams was supposed to have: that way it would be easier, in exam conditions, to know whether or not you'd missed something out. At the time I think I was rather shocked that a subject could require you to treat its content so cynically. In retrospect it was probably perfectly good advice: if you like biology, and you want to study it further, then it makes sense to use a bit of cunning in order to get through the more elementary material. I couldn't understand how it would happen that someone would decide that they liked biology; to some extent I still can't.

The closest I've ever come to acquiring a taste for this sort of thing was with the smattering of cognitive neuropsychology included in my undergraduate degree. I recall enjoying the difficulty of it: the careful thought needed to see what it was that some theory was predicting or that some evidence was showing. Badiou is even more damning here: “Even more so than biology, [cognitive science] is just a mass of facts and techniques, devoid of concepts or adequate formalisms […] It is no more advanced in its understanding of the phenomena than was Gall's phrenology.”

At university I was particularly struck by the theorising about implicit learning and memory: the idea that it is possible for someone to learn something without being able to give an account of their knowledge, or even being able to remember having learnt. It is apparently possible for amnesiacs to acquire new skills. In 19^th century psychology it was thought that the brain must contain memory traces that could influence behaviour but were either too weak to reach consciousness or had become dissociated from the ego. But in the 20^th century it became generally accepted that some memory must differ in kind from that normally considered. It seems that this starts with Henri Bergson, of all people: “The past survives under two distinct forms: first, in motor mechanisms; secondly, in independent recollections.” (See Schacter, D. L. (1987). “Implicit memory: history and current status.“ J of Exp Psych: LMC, 13 for more details.)

This, of course, can be of no importance for Badiou: if truth is to be distinguished from mere correct judgement, then it cannot be understood by looking at the mechanisms of knowledge (be they neural or institutional). Nonetheless, it can't be completely without interest. The distinction between two different kinds of memory, as well as being noted by Bergson, shows up in 20^th English philosophy as the difference between ‘knowing that’ and ‘knowing how’, and loosely corresponds to ‘present at hand’ vs ‘ready to hand’ in Heidegger. Might this not qualify it as being a decisive event in the history of philosophy? (My knowledge of Bergson and Heidegger is vague and largely second-hand—but my understanding is that they treated this distinction as being much more than some arbitrary psychological fact.)

Tangency

Having just suggested that Descartes was not important as a mathematician, I thought I'd try to back that up via a consideration of the influence of his work on the invention of the calculus.

When calculus is taught, differentiation is commonly the first aspect of it to be covered, and this is in terms of it being a method for finding tangents to algebraic curves. Prior to the invention of the calculus there were other ways—less general and more difficult—for finding tangents. One of these was given by Descartes in La Géométrie. This depended on an insight that was partially geometric, and partially algebraic.

Suppose you have a curve, and a point on the curve at which you want to find the tangent. Now suppose there is a circle that passes through this point. The circle will probably cross from one side of the curve to the other, and then cross back again somewhere else, so intersecting with the curve twice. However, if the circle just touches the curve once then the tangent of the circle at this point will be the same as that of the curve, and hence (perpendicular to this) the normal of the circle will be the same as that of the curve. The centre of the circle will lie on this normal.

In algebraic terms, the intersections between the curve and the circle will be given by roots of some equation. If there are two intersections then there will be two roots. However, if the circle just touches the curve, and so there is only one intersection, then this corresponds to the equation having a repeated root. [This is like, for example, the equation x²-2x+1=0, which can be factorised as (x-1)(x-1)=0, having a repeated root at x=1.]

Putting this together: given a curve and a point, if you can contrive a way to find a circle that passes through the point, and which gives a repeated root in the equation describing how it intersects with the curve, then you can find the tangent to the curve as being perpendicular to the line passing through the point and the centre of the circle.

Applying this method wasn't terribly straightforward. However, Jan Hudde devised a refinement to this method that made it easier to find a suitable circle. As far as I can tell (i.e. going by MA290 TV5) it went something like this:—

Suppose, for example, that you want to find the tangent to the curve y=x³ for some particular value of x. Let there be a circle with its centre on x-axis at (c,0), and which cuts the curve at the point you want the tangent, and also somewhere else. Call the difference in ordinates of the two intersections e:

The first step is to find an equation relating e to c. The intersections are at (x,x³) and (x+e,(x+e)³). And as they are both on the circle, their respective distances from (c,0) must be the same. This gives

(x+e–c)² + ((x+e)³)² = x² + (x³)²

Expanding this out and then simplifying gives

2e(x–c) + e² + 6x⁵e + 15x⁴e² + 20x³e³ + 15x²e⁴ + 6xe⁵ + e⁶ = 0

i.e.

2e(x–c) + 6x⁵e + e²f = 0

where f=1+15x⁴+20x³e+15x²e²+6xe³+e⁴.

Then dividing through by e and re-arranging a bit gives

x–c = –3x⁵ - ef/2

The point of all of this is so that we can now apply one of Fermat's tricks. If the second intersection ‘adequates’ to the first then the circle be just touching the curve:

Also e, the difference between them, disappears. This gives

x–c = –3x⁵

So the slope of the normal is x³/(x–c)=–1/(3x²), and hence the slope of the tangent is 3x² (as you might have expected).

As I recall, Isaac Newton's initial work on what came to be called differentiation was done in 1665 when he was working through this stuff from his copy of La Géométrie. Instead of going from two adequal points on a curve to a circle passing through them, and then (via its normal) to the tangent, Newton realised that you could get the tangent from the chord joining the two points.

In the example above, the slope of the chord is

((x+e)³ – x³) / e
= (x³ + 3x²e + 3xe² + e³ – x³) / e
= 3x² + 3xe + e²

so as e disappears this becomes 3x².

Whilst it might appear that Newton was building on Descartes' work, closer inspection of the developments shows a different picture:

Fermat had developed ways of reasoning about 'adequal' quantities that he had applied to tangency problems, although he did not develop his ideas very far. More importantly, he had not published them. However, they were presumably known amongst Mersenne's circle of correspondents.
Descartes devised a different way of dealing with tangency problems which he did publish, but which was very cumbersome to apply.
Hudde used Fermat's ideas to simplify Descartes' method. This was included in the later editions of La Géométrie.
In his reading of La Géométrie, Newton extracted Fermat's ideas from Hudde's method. He generalised them, whilst excising Descartes' one contribution to the solution of tangency problems.

This is why the ‘Descartes effect’ in early modern mathematics cannot really be understood simply in terms of Descartes' own work as a mathematician.

TMA 2.2

Question:

To what extent is it fair to claim that mathematics developed in the sixteenth and seventeenth centuries primarily as a means for responding to technological and scientific needs?

The influence of science and technology on mathematical development in the 16^th and 17^th centuries

The 16^th and 17^th centuries were times of great change in Europe: the Reformation; the impact of printing; exploration of the New World; loosening of scholastic dogmas; the rising urban middle class; the development of technology and the invention of modern science. Specific to mathematical history there was the ongoing recovery of ancient Greek works; an increase in vernacular texts; innovations in symbolism; the emergence of an pan-European mathematical community; developments in algebra and its new use in geometry. These things were bound up with each other in complex ways; when it comes to mathematical development over the course of these two centuries there is no reason to think that the needs of technology and science were its sole cause.

A clear-cut case of new mathematics being developed due to such needs is that of Napier’s logarithms, which were invented and refined with the explicit aim of easing the tiresome calculations needed in many applications of mathematics. However, it is also possible to point to important mathematics developed without application. The researches in algebra in 16^th century Italy are such: no problem in commercial arithmetic would require cubics for their solution. Later algebraic work, such as Fermat’s on number theory, can similarly be regarded as pure research.

The requirements of technology and science are only likely to result in new mathematical developments if the requirements themselves are new. In some cases this was true: Kepler’s computationally intensive work was made possible by increasing accuracy in astronomical observation; and the investigation of spherical geometry in relation to navigation became important to the various European nations that were developing their interests in the New World (and so had to negotiate the Atlantic). However, some important practical requirements were not new: in surveying there was a tradition of “sub-Euclidean” geometry going back through the Middle Ages to Roman agrimensores; and so, after the development of trigonometry in the 15^th century, it was not the creation of new knowledge that was important so much as the changing distribution of existing knowledge.

Knowledge was propagated in the form of printed manuals for literate artisans or embodied in the use of instruments. In itself this is not a properly mathematical development, but is important nonetheless. If nothing else, the increasing importance of mathematical knowledge for diverse practical applications resulted in improvements in its teaching: a tide on which all boats rise.

Mathematical developments certainly made some scientific advances possible. Newton’s mathematical physics would have been inconceivable without the algebraic approach to geometry instigated by Descartes. However, this does not make the science the cause of the mathematics: Descartes was not motivated by a desire to produce such a physics, and his own physics was not mathematical in this sense. For Descartes mathematics was a source of example problems which could be used to demonstrate his method and to sharpen his logic. In his philosophical schema this logic provided the basis for his metaphysics which in turn founded his physics.

In this example it is plausible that the emergence of algebraic geometry was in part due to styles of study arising from the invention of printing. Learning became more book-based, and—as Peter Ramus noted—the synthetic proofs of classical geometry did not produce much insight in themselves. Moreover, the distribution of books enabled scholars to work outside of traditional institutions; but, again, the use of synthesis in Greek works did not furnish them with a method of invention. Using algebra resolves this by allowing wholly analytic solutions: a problem could be assumed solved—as normal—but the unknowns of this solution could be represented symbolically, manipulated, and solved in the course of analysis. This made it ideal for independent scholars like Descartes. Newton, although belonging to a Cambridge college, seems to have largely taught himself from books: hence his early enthusiasm for Descartes’ Geometry over Euclid’s Elements.

Pure maths research and practical applications could be closely related. Two contrasting examples of this are provided by calculus and projective geometry. Calculus originated in the theoretical treatment of tangents and areas of curves, in the work of mathematicians such as Fermat and Cavelieri; and Newton brought this work to fruition in order to provide a basis for his work in physics. Whereas the origins of projective geometry were practical, in the theory of perspective drawing created by Leon Battista Alberti in the 15^th century; but Desargues, who wrote a manual of perspective, used this as a starting point for his theoretical endeavours.

It is conspicuous that much mathematical thought had a mechanistic flavour: Napier conceived of his logarithms as being generated by concurrent motions, and Newton thought of tangents to curves in terms of instantaneous direction of motion. Although this was not universal—Kepler’s account of logarithms and Descartes earlier treatment of tangency were both statically geometric—such a conceptualisation was not considered disreputable. Similarly, Kepler’s and Galileo’s view of the world as inherently mathematical meant that—for all the avowed Platonism—physics was not a separate field of investigation.

Overall, it is too simplistic to say that practical needs drove all the mathematical developments in this period. It is true that it drove some, but it is also true that the influence could be much less direct than this thesis suggests, and also that technology, science and mathematics were all influenced by other aspects of these changing times.

Thoughts

I'm reasonably happy with that, although there are certain things in it that I now think are wrong. I don't even think I could have had much of a problem with the word count (otherwise I can't see why I would have given Alberti's name in full).

I think I over-estimated the important of Descartes (easily done) in a way that tends to undermine my comments about institutional changes in the period. I also think I missed a big bit of the picture about the invention of the calculus, and consequently the influence of applied maths was more pervasive that I realised.

The comment about “the algebraic approach to geometry instigated by Descartes” is misleading if not outright nonsense. The phrasing of it comes about because I knew that François Viète's work predated Descartes' (by about half a century), but that Descartes' approach was significantly different, chiefly in its essentially dimensionless treatment of magnitudes. (For example, for Descartes two lengths could be multiplied to give a third length; whereas for Viète they would give an area that could not be compared directly with lengths, only with other areas.) I am now not convinced by this, and am more inclined to believe John Wallis' famous assessment that there was little or nothing in Descartes that wasn't to be found in Viète, William Oughtred or Thomas Harriot.

There were lively algebraic traditions in England and the Netherlands, and—as far as I can see—the main reasons for Descartes' standing as a mathematician was that he was treated as a figurehead by certain Dutch mathematicians (van Schooten, Hudde, van Heurat) and that the ‘commentaries’ included in the later editions of Descartes' Geometry were used as a vehicle for propagating their research results. (With the third edition of 1683, only a quarter was actually by Descartes.) But for that it is conceivable that Descartes' views on geometry would have had of no greater impact that Bishop Berkeley's views on the calculus.

The reason that this undermines my point about universities vs independent scholars is that the Dutch mathematicians were working in the context of a university system. I still think I had a point—apparently John Wallis taught himself mathematics from Oughtred's Clavis Mathematicae—but I didn't really substantiate it.

I said that my picture of build-up to the calculus was decificient. I won't go into detail here, but it seems likely that development of numerical methods, particularly of interpolation techniques—used in the production of trigonometric tables required in navigation and astronomy—formed an important part of the calculus' conceptual basis.

TMA 2.1

This was a gobbet question concerning an extract from John Dee’s preface to the first Engish translation of Euclid’s elements mathematics. I won’t quote it, as it is quite long.

John Dee on navigation

The extract is taken from John Dee’s Mathematicall praeface to Henry Billingsley’s English translation of Euclid’s Elements of 1570. In this preface (together with his ‘Groundplatt’ diagram) he is giving a schematic overview of the mathematical sciences, both pure (“Principall”) and applied (“Derivative”). In the extract he is discussing navigation in relation to mathematics. In the course of the extract he gives a definition of the art of navigation (finding sailing routes between arbitrary places, and determining position at sea); he relates this art to various parts of mathematics; he gives a long list of instruments used in navigation; and he argues for the particular importance of navigation to England as an island nation.

In discussing the arts pertaining to navigation he is indicating what knowledge a master pilot—as a practitioner—needs. He gives four specific arts (“Hydrographie, Astronomie, Astrologie, and Horometrie”), these being founded on the “Principall” sciences of geometry and arithmetic. The instruments he lists are those a pilot should be able to use (and make).

Of these arts, hydrography relates to plotting routes and understanding charts; with the attendant difficulties of considering spherical geometry whilst dealing with flat maps. The abilities to make astronomical observations and to reckon time would be used at sea in determining a ship’s position and bearing. It is not clear why Dee includes astrology in the list: Dee’s sympathies in this area are well known, but he does not expand on the topic. It is possible that he meant nothing more than knowledge of the constellations and the movements of the planets.

Billingsley’s translation made Euclid accessible to unlatined English-speakers for the first time. And for Dee to emphasise the usefulness of mathematical understanding puts him in the tradition of Robert Recorde, who would put similar ‘pitches’ to his potential readers in his vernacular mathematical works.

The book, and Dee’s preface, was not specifically aimed at navigational practitioners: The Elements is a work of pure geometry. Dee makes it clear that geometry, together with arithmetic, is the foundation of all the navigational arts, but does not indicate to what extent pilots should study its principles. Perhaps Dee’s intended audience here consists of those who might study geometry in order to further its application, continuing Dee’s work in devising useful arts for practitioners to learn.

Dee was an ardent admirer of the ancients, and so it is fitting that he would choose a translation of the most famous of Greek mathematical texts as the vehicle for this mathematical manifesto. The recovery and publication of Greek texts had been the object of programmatic effort in Italy around this time. (Francesco Maurocilo was a little earlier. Federigo Commandino was a contemporary with whom Dee had corresponded concerning a possible lost work of Euclid.) The motivation was the perceived superiority of the ancients over Islamic and mediaeval authors. This program included Italian translations, as Dee notes in his preface, although Commandino’s Italian version of The Elements was not published until 1575.

Dee’s preface is not particularly concerned with promoting the content of one particular book, as being either profitable for practitioners or edifying for scholars. He is proposing a systematization of all mathematics that (as shown on his Groundplatt) would encompass both theoretical, Euclidean geometry and the practical arts of navigation (and much else besides). His vision could be taken as pedagogical reform in the manner of his friend Peter Ramus: the Groundplatt is a ‘dialectic’ presentation such as Ramus might produce, and is described by Dee in a Ramist turn of phrase as being “somewhat Methodically contrived”. It could also be taken as the outline of a program of research, such as the ones Descartes and Frances Bacon would later produce.

Navigation was an increasingly important matter in England at this time, with the growth in trade and the exploration of the New World, as Dee suggests in this extract. He mentions devices he invented for the Moscovy Company, a trading company to whom Dee was a technical advisor. Recorde had previously acted in a similar capacity, writing instructional books for their navigators’ use. Works that follow up Dee’s remarks on navigation in the preface included William Bourne’s A regiment for the sea and Dee’s own General and rare memorials pertayning to the perfect arte of navigation. Dee’s friend Mercator devised a projection most suitable for navigational charts, the basis of which was later worked out by Edward Wright and Thomas Harriot.

It is appropriate that Dee should include a list of instruments in his account of navigational knowledge. Practical mathematics is primarily carried out by various sorts of practitioner, rather than by scholars, and instruments were an important part of this. Certain trades had always had their instruments, but around this period there was an increased inventiveness. This became more pronounced slightly later, with more mathematically sophisticated instruments devised by people such as Edmund Gunter of Gresham College.

Dee is well known as a sympathiser of hermetic philosophy, a form of neoplatonism common in the early Renaissance. Dee’s work on navigation exemplifies the fact that hermeticism—unlike Platonism proper—tended not denigrate worldly matters, and that it emphasised the power that knowledge brings.

Thoughts

Oh dear, this is very flat, isn’t it? I think this was my first attempt to write methodically: making sure I had things to say about all the things that needed them (author, target audience, relevant mathematical traditions, etc) and then just stringing them together. It might be inevitable that an answer to this sort of question will be a sequence of facts and observations that lacks any real point.

And I do bang on about mathematical practice and practitioners. There’s one learning outcome that I’ve definitely demonstrated. (Not that MA290 was like that.)

These days Dee is mainly remembered for conjuring spirits with Edward Kelley. I still find it surprising that in his day he was a key figure in English mathematics. It is even more surprising that he was friends with Peter Ramus, scourge of all dunsicality (and called “the pedant of France” by Giordano Bruno). Even though I should know better, I find it hard not to think in terms of an anachronistic distinction between respectable science and occultism. (Although I recall that Ramists were, for example, against the teaching of astrology, but not because it was diabolical, or even because it was false, but because in their classification it was a ‘specific’ rather than a ‘general’ art.)

I have a vague theory about the different perception of John Dee and Isaac Newton. At some point, presumably during 18^th century, the connection between early modern science and occultism became an embarrassment. Because Dee’s occult interests were well known his importance as a mathematician had to be played down. Whereas Newton was firmly at the heart of the scientific pantheon, so his interest in alchemy had to be treated as peripheral. (There’s a similar story with Johannes Kepler.)

[At this point I was going to go on about the relative obscurity of Thomas Harriot being less due to his association with the “School of Night” than his failure to publish on his most important work, but I'm getting off topic.]

Quines

I've never read ‘Gödel, Escher, Bach’. (I've been lent it on occasion, but I never took to it.) Consequently, I only vaguely know about quines.

A quine is a program that produces its own source code as output. Such a program will contain a representation of the program. However, this representation is part of the program, and so must itself be represented. This is where it gets a bit fiddly.

Quines had always seemed a bit mysterious, but I've just read David Madore's account and it turns out that they are almost disappointingly straightforward.

The trouble is that quines I've seen have often been difficult to read. There are two reasons for this. One is that they are usually very terse. (The shorter a quine's source code is, the less work it has to do in producing it: there is some advantage in having the source consist of a single line.) The other is that a typical quine will contain text data that is very similar to the rest of the program: this doubling can often be visually confusing. Both of these factors are evident in this example.

I've tried to write a quine that avoids both of these problems.

#!/usr/bin/perl

$d='
H4sIAG+KyEcAA1WPUUvDMBSF3/Mrrl3LEgidW0vElL4oOvYwffDNMWQ1cQba
puQmyPz1piuD+XI598C593yzm0VAt2hMvxhO/tv2hKg6SZIM4yAkQ9MN1nlo
DqhFyeHNO9MfN68cjr9m4IAnJAHq85av43g2raYvttccEpdwmOQlll8EnQ7m
jSiV/rRKU8UYy50+KMpIvJqjVzb4/McZr2nYLe/v5Gol9pABVRzCLi6yEGLP
GJldQWjXEpKqep7hvBoBAmrYbrZPUj6cX1ZkdB5tNziNKOV7a5qKpCPEf1PK
Tnfr0EcoOnX8mErTNHatyBBJ/BdgaNA7mgYOtxyKgnFIY8Eruyg4LEUZI38o
ePPObgEAAA==
';

use MIME::Base64;
use Compress::Zlib;
$u = Compress::Zlib::memGunzip(decode_base64($d));
printf substr($u, 0, 33), $d, substr($u, 33, 164);

In a quine there are typically two important parts: some data, and some logic which decodes this data. In my program, the data is in the variable $d, and last 4 lines are the decoder.

The data is base64 encoded: partly to disguise its content (avoiding the 'doubling' problem), and partly as it means that the data won't have any special characters (quotes, backslashes, etc) that would complicate the decoder. The important thing is that the four lines of the decoder are stored within it.

The decoder has to print out the whole program: both data and decoder. This means it has to process the data twice over: once ‘raw’ and once decoded. This is why the decoder prints both $d itself, and substr($u, 33, 164)—which is derived from the data in $d. In writing this program I used a ‘skeleton’ version, where the data was missing. I wrote another program to work out what the correct data should be.

To tie everything together there is also a formatting string. For convenience I've also stored that within the data in $d.

Is that really all there is to it?

TMA 1.2

"Consider the problem of doubling the cube. Explain what it is, and why it was important in the development of mathematics during Greek times."

Cube duplication in Greek mathematics

The problem of cube duplication is that of constructing a cube with exactly double the volume of a given cube. It is equivalent to the problem of doubling the volume of an arbitrary solid figure whilst maintaining its proportions. An early result, attributed to Hippocrates of Chios [1], was that the problem could be solved if it were possible to construct two 'mean proportionals' for two given lengths[2]. Given the side of the original cube, a line double this, and two mean proportionals between, the first proportional is the side for the doubled cube. It was in this 'reduced' form that the problem was subsequently studied.

A traditional account of the problem's origin is given by Theon[3]:

when the god announced to the Delians by oracle that to get rid of a plague they must construct an altar double the existing one, their craftsmen fell into a great perplexity in trying to find how a solid could be made double of another solid, and they went to ask Plato about it.

Of course, the problem must have been know previously if Hippocrates had worked on it a century earlier. However, a number of geometers associated with Plato did work on the problem.[4]

Even without this story the problem is intelligible in the context of the Greek geometric research tradition. It is analogous to the planar problem of doubling a square, the solution of which appears in one of Plato's Socratic dialogs[5]. A letter, supposedly by Eratosthenes and quoted by Eutocius[6], indicates that the solution of the problem of two mean proportionals would allow the cubature of many other solid figures. This is analogous to the relationship between the quadrature of rectilinear, planar figures and the finding of single mean proportionals.[7] The letter also states that a solution allows for arbitrary scaling, not just doubling.

This letter also suggests a practical significance. Scaling solid figures correctly is useful in building, and also in the construction of war engines. The cubature of solid figures helps in the manipulation of measures of volume. This slant is reinforced by a criticism that earlier researchers had produced theoretical solutions but not practical constructions. (The solution given in the letter is the design of a mechanical device for finding mean proportionals.)

Proclus credits Hippocrates with being the first to use reduction as a technique in difficult constructions. It was later used in other areas.[8] A reduction is, in a sense, an uncompleted analysis. It has been suggested[9] that this reduction was via a generalisation of the problem to the cubature of a square-based cuboid, and was analogous to a lemma concerning cylinders in a proposition of Archimedes.[10]

Eutocius' commentary on this proposition is an important source on Greek research in this area. He gives a collection of solutions for the reduced problem, using a variety of construction means: no line-and-circle solutions[11]. The mechanical flavour of the solutions sits ill with Plato's notion of geometry. One solution due to Menaechmus, a young contemporary of Plato, involved the intersection of a parabola and hyperbola: curves produced by planar sections of a cone.[12]

The Greek interest in conic sections appears to start with Menaechmus' solution. Conics was an important topic in later Greek mathematics, reaching a state of considerable sophistication with Apollonius around a century later.

The importance of conics is indicated by a three-fold classification of geometric problems given by Pappus[13]: plane (soluble by line and circle), solid (needing conic sections), linear (needing more exotic curves). He says of "ancient geometers"[14]:

they were as yet unfamiliar with the conic sections and were baffled for that reason

The significance of the problem is two-fold. One is the use of reduction, which allow problems to be investigated and understood in relation to other problems, even if they were left unsolved. The other is that this problem[15] pushed Greek geometry beyond the confines of line-and-circle constructions: specifically, it initiated the study of conics, which became an important topic in its own right.

Footnotes

[1] By Proclus. See SB2.F2
[2] Which is to say, two lines of intermediate lengths such that the following ratios are all equal: between the first given line and the first proportional, between the second proportional and the second given line, and between the proportionals themselves.
[3] Theon credits Eratosthenes as his source. See SB2.F1
[4] The version of the story by Eutocius mentions Archytas, Eudoxus, and Menaechmus
[5] In Meno; see SB2.E1
[6] See SB2.F3
[7] For more details see Saito (1995)
[8] e.g. Archimedes reduction of the quadrature of the circle to the rectification of its circumference
[9] Saito (1995)
[10] Proposition II.1 of On the sphere and the circle; see SB4.A6.
[11] Which are now know would be impossible
[12] Other given solutions included a construction involving intersecting half-cylinders from Archytus, a neusis construction from Heron, and a use by Nicomedes of a curve called a conchoid. See Thomas (1991).
[13] See SB5.B4.
[14] Strictly speaking he is referring to another problem— angle trisection—but the point applies to any problem that is ”by its nature a solid problem“. Pappus also criticises the use of neusis constructions for solid problem: these are only necessary for linear problems.
[15] Together with the other 'classic' problems: circle quadrature and angle trisection

Additional References

Ken Saito, “Doubling the Cube, A New Interpretation of its Significance in Early Greek Geometry”, Historia Mathematica 22, 119-137
Ivor Thomas (translator), 1991, Greek mathematic works I - Thales to Euclid, Harvard University Press

Thoughts

My main recollection from this was that I had great difficulty keeping within the word limit. I had the idea that quotations and footnotes shouldn't count. Discounting quotations meant that if I could find some primary source saying something I needed said then I could just use that. I discounted footnotes mainly so that I could demonstrate that I knew what 'mean proportionals' are without wasting words on it. (I tried to come up with a compact definition, but couldn't devise anything really suitable.) Understandably, my tutor didn't hold with my use of footnotes.

At the time I thought that there ought to be more to say about Pappus' classification of geometric problems, but I now can't quite remember what. Part of it might have been that it was never clear to me quite why solutions involving conic sections were considered more acceptable than neusis constructions. It also might have been to do with an analogy with Descartes' later classification of curves (wherein he used a tenuous notion of a curve's 'simplicity' in order to exclude from geometry figures for which his method did not work).

TMA 1.1

This was a gobbet question concerning a problem from ancient Egyptian mathematics.

‘SB’ indicates a reference to the source book, i.e. Fauvel and Gray, and “SB1.D1” means extract D1 from chapter 1. In this case the extract comes from A. B. Chace's 1927 translation of the Rhind Papyrus.

A quantity and its 1/2 added together become 16. What is the quantity?
Assume 2.

\ 1 2

\ 1/2 1

Total 3.

As many times as 3 must be multiplied to give 16, so many times 2 must be multiplied to give the required number.

\ 1 3

2 6

\ 4 12

2/3 2

\ 1/3 1

Total 5 1/3.

1 5 1/3

2 10 2/3

Do it thus:

The quantity is 10 2/3

1/2 5 1/3

Total 16.

(untitled)

Problem 25 of the Rhind Papyrus consists of the statement of a numerical problem requiring an unknown quantity to be determined; a series of calculations that arrive at an answer; and a final calculation demonstrating that the value found satisfies the problem statement. It is very similar in form—both the nature of the problem and the method of its solution—to Problem 24 (SB1.D1).

The problem to be solved is stated as: “A quantity and its 1/2 together become 16. What is the quantity?” In modern terms we might call the quantity x, and say “x + 1/2 x = 16. What is x?”.

A modern way of solving this would be to manipulate the equation until it gave an answer: re-expressing x + 1/2 x as 1 1/2 x, and then dividing both sides by 1 1/2. However, the approach taken by the scribe is to make a “guess” at the answer (2), and then use a comparison between the total yielded by the guess and the one given by the question in order to adjust the guess and so get the correct answer (10 2/3).

To describe the approach in modern terms we might call the guess x′, and say that the scribe relies on

(x + 1/2 x) ÷ (x′ + 1/2 x′)

x ÷ x′

being true regardless of the value of x′ (provided it is not zero); and so

x′ × (x + 1/2 x) ÷ (x′ + 1/2 x′).

Given that x + 1/2 x = 16, and choosing x′ = 2 we get

x	=	2 × 16 ÷ (x′ + 1/2 x′)
	=	2 × 16 ÷ 3
	=	2 × 5 1/3
	=	10 2/3.

The scribe performs the calculations in these lines (using the normal Egyptian method for multiplication and division) in the order given. The principle I said the scribe relied upon is expressed: “As many times as 3 must be multiplied to give 16, so many times 2 must be multiplied to give the required number.”

Assuming 2 is not a “good guess” in the sense of being close: the method doesn't require this. It looks rather that it was chosen so that the total given by the first step of the calculation is an integer. Dividing by a non-integer would have been cumbersome using Egyptian notation for fractions: had this been easy the problem could have been solved more directly.

The final check most likely served to guard against simple mistakes in the calculations, and maybe also against the wrong method being used.

Chace (SB1.D6) and Toomer (SB1.D7) differ as to whether the Rhind Papyrus shows mathematics pursued for its own sake or is a textbook for scribes. Problem 25 looks more like a textbook than research. The problem is rather easy: possibly too easy to justify the method used for its solution; certainly too easy to require that every step of the calculation be set down. Rather, it is the method of solution that is the point of the text, and the particular problem given is being used to illustrate it. (The final check illustrates good practice.)

The text shows that the Egyptians were comfortable with number as an abstract notion: the problem deals in quantities, but the text does not specify of what; and the intermediate result 5 1/3 is dimensionless. More abstractly still, the scribe was able to consider a quantity without knowing what is was.

However, although this text seems to be for explaining a method for solving a whole class of problem, it does so purely in terms of a particular example. There is no evidence that the scribe could have described the method being used in general terms, let alone investigate its validity and scope of applicability

This limited level of abstraction fits with the relative lack of development in Egyptian mathematics. The Egyptians could devise and codify fairly abstract methods for solving concrete problems, but did not take the next step of pursuing abstract problems.

Thoughts

There's not very much to say here. At the time I did wonder if my account of why the Egyptian method worked was a bit unnecessary: however, I'd decided (rather arbitrarily) that equations shouldn't be included in the word count, so cutting that section wouldn't have given me much scope for doing anything else extra.

The method used was later called “the rule of single false position”, with there being a rule of double false position for slightly more complicated problems. ‘Later’ meaning the Middle Ages, when these terms appeared in treatises on commercial arithmetic (such as the works of, uh, at least one of Fibonacci and Luca Paccioli). The fact that this method had a name implies that it was still in use then, although I don't know how long it continued to be used. I suspect that double false position would have had a longer life than single false position. I know that double false position was described in Robert Record's popular English arithmetic The Ground of Artes of 1543.

Both single and double false position can both be thought of in terms of a straight-line graph, where the question is to find an input that will produce a give output. The problems soluble with single false position are the ones where this line passes through the origin, which is why only one guess needs to be made. Double false position can cope with problems where this is not the case, but requires two guesses to be made.

The main benefit of single false position is that it avoids dividing by a fraction; so I imagine that it would lose its appeal as reasonable notation for vulgar fractions exists. Whereas double false position avoids some mildly more involved algebraic re-arrangement, and so might continue to be convenient even when other ways of solving such problems are available.

MA290

I've recently completed my first Open University course: MA20 Topics in the History of Mathematics. I remember watching some of the TV programmes for this—which are supplied on DVD these days—when I was a kid. I had already picked up the course book—The history of mathematics: a reader eds. John Fauvel and Jeremy Gray— a number of years ago in a book sale, and so when I found at that this was the last year that the course was running I really had to sign up.

I found doing the assignments (TMAs) an interesting experience. I've always found writing—even short things—a fairly painful process. Here the problem was compounded rather by the assignments having word limits. However, there were points where I thought I was getting the hang of it, and I started to get a sense of the meaning of the word ‘copy’ as a mass noun: not to do with quantity of words so much as a certain detachment, slabs of verbiage to be shoved around.

I'm going to put up my TMA answers here, as I want to mull over a couple of them some more.

Conics

I don't think it's normally a good idea to try to translate something on a topic you don't understand from a language you don't speak. But that's OK: I like a challenge. So, here's a tiny bit of La Hire's Nouvelle Method.

Defintion
I'appelle une ligne droitte A D couppée en 3 parties harmoniquement quand le rectangle contenu sous la toutte A D & la partie du milieu B C est égal au rectangle contenu sous les deux parties extreme A B, C D: ou bien lorsque la toutte A D est à l'une des extremes A B ou C D comme l'autre extreme C D ou A B est à la partie du milieu ce qui est la mesme chose.

I say that a straight line AD is cut harmonically into 3 parts if the rectangle contain by the whole AD and the middle part BC is equal to the rectangle contained by the two extreme parts AB and CD; or equally, that the whole AD is to one of the extremes AB or CD as the other extreme CD or AB is to the middle part, which is the same thing.

Well that's not too bad.

Lemme 1 (Fig 1)
Coupper une ligne droitte donnée A D en trois parties harmoniquement.
De l'une des extremitez D de la ligne A D soit tiré la ligne D G faisant quelqu'angle avec la ligne A D & soit D G à D E en quelle proportion l'on voudra, & ayant tiré la ligne G A, par la point E on menera une ligne E C F paralelle à G A & C F estant prise égale à C E que l'on joigne G F qui couppera la ligne A D au point B: Je dis que comme D A est à D C ainsi B A est à B C.
Dans le triangle D G A la ligne E C étant paralelle à la base G A, D G sera à D E comme D A à D C & comme D A à D C aussi G A à E C ou à C F son égale: mais à cause que les lignes G A & C F sont paralelles les triangles B A G, B C F seront semblables, & par consequent G A sera à C F comme B A à B C, mais G A est à C F comme D A à D C: D A sera donc à D C comme B A à B C, ce qu'il falloit faire.

To cut a given straight line AD harmonically into 3 parts.
From one of the extremities D of the line AD draw the line DG making some angle with the line AD, and make DG to DE in whatever proportion you want. Having drawn the line GA, from the point E take a line ECF parallel to GA and such that CF equals CE. Draw the line GF which cuts the line AD at point B. I say that DA is to DC as BA is to BC.
In the triangle DGA the line EC is parallel to the base GA. DG is to DE as DA is to DC, and DA to DC as GA to EC or to CF (its equal). As the lines GA and CG are parallel the triangles BAG and BCF are similar, and so GA is to CF as BA to BC. But GA is to CF as DA to DC. Therefore DA is to DC as BA to BC. QEF

I think there's something a little rum about his use of colons. Also, I have no idea what tenses the various verbs are in (e.g. sont vs seront).

Lemme 2 (Fig 2)
Si une ligne droitte A D estant couppée en trois parties harmoniquement, & ayant pris un point E hors de cette ligne mesme se elle étoit prolongée, si l'on tire de ce point E des lignes prolongée par les poins de division A, B, C, D de la ligne A D: Je dis que la ligne F I menée paralelle à A D & couppant les 4 lignes E A, E B, E C, E D aux poins F, G, H I sera aussi couppée en ces poins en 3 parties harmoniquement.
Dans le triangle E A D la ligne droitte F I est paralelle à la base A D, donc dans chaque triangle E A B, E B C, E C D les parties de la ligne F I à sçavoir F G, G H, H I seront paralelles aux bases A B, B C, C D; c'est pourquoy comme A D est à F I ainsi E A est à E F & comme E A est à E F ainsi A B est à F G: Mais comme E A est à E F ainsi E B est à E G: Mais comme E B est à E G ainse B C à G H. deplus comme E A est à E F ainsi E C est à E H; & comme E C est à E H ainsi C D à H I, c'est pourquoy comme la toutte A D & chacune de ses parties A B, B C, C D, sont entr'elles ainsi la toutte F I & chacune de ses parties aussi F G, G H, G I seront entr'elles estant chacune separement l'une à l'autre comme E A à E F ainsi qu'il a esté démontré. C'est pourquoy puisque A D est à A B, comme C D est à C B; ainsi F I sera à F G comme H I à H G, ce qu'il falloit prouver.

A straight line AD is cut harmonically into three parts: take a point E not lying on this line (even if it is extended), and extend lines from this point C to the points of division A, B, C, D of the line AD. I say that a line FI taken parallel and cutting the 4 line EA, EB, EC, ED at points F, G, H, I will also be cut harmonically into 3 parts at these points.
In the triangle EAD the straight line FI is parallel to the base AD, hence in each triangle EAB, EBC, ECD the parts of the line FI (to wit, FG, GH, HI) are parallel to the bases AB, BC, CD; this is why as AD is to FI so EA is to EF, and EA is to EF as AB is to FG. As EA is to EF so EB is to EG. As EB is to EG so BC to GH. Moreover, as EA to EF so is EC to EH; and as EC to EH also CD to HI, which is why as the proportions amongst the whole AD and each of its parts AB, BC, CD are, so the proportions amongst the whole FI and each of its part FG, GH, GI will be, as is shown. This is why just as AD is to AB, as CD is to CB; so FI will be to FG, as HI to HG. QED

I got a bit lost toward the end of the second paragraph here. Having all the proportions spelt out in words isn't helping at all. It also seems to me that mais is being used as a generic connective. I don't know what to make of sçavoir.

Scholie
Mais si l'on tire par les poins de division A, B, C, D, de la ligne A D des lignes E A, E B, E C, E D paralelles entre'elles: Je dis de mesme que la ligne F I menée paralelle à A D couppant ces quatre lignes aux poins F, G, H, I, sera divisée par ces mesmes poins en 3 parties harmoniquement.
La demonstration en est évidente puisque ces 4 lignes E A, E B, E C, E D étant paralelles entr'elles & les 2 A B, F I l'étant aussi entr'elles composent les 4 paralellogrammes A I, A G, B H & C I & pat consequent les costez opposes seront égaux & en mesme proportion entr'eux, ce qu'il falloit démontrer.

Take from the points of division of the line AD some mutually parallel lines EA, EB, EC. I say that the line FI taken parallel to AD cutting these four lines in points F, G, H, I will be divided harmonically into 3 parts by these points.
The proof is evident as these 4 lines EA, EB, EC & ED—being mutually parallel—and the 2 lines AB & FI—also being mutually parallel—form the 4 parallelograms AI, AG, BH & CI and so the opposite sides will be equal and in the same proportions. QED

He doesn't say as much, but he's taking E to be a point at infinity here. It wouldn't change the argument if the 4 parallel lines were designated without making this step.

Lemme 3 (Fig 3)
Les mesmes choses que cy-devant étant posée: si l'on mene la ligne droitte F H paralelle à l'une des extremes E A ou E D des 4 lignes menées du point E par les poins de devision de la ligne A D: Je dis que la ligne F G H sera couppée en 2 parties égales par les 3 autres lignes E A, E B, E C.
Du point F on tirera la ligne F d paralelle à A D & du point H on tirera H I paralelle à celle du milieu E B des trois lignes, qui couppent la ligne F H jusques à la rencontre de F d en I.
Par le Lemme precedent la ligne F d sera couppée en 3 parties aux poins F, c, b, d harmoniquement: mais à cause des paralelles E d & F G les triangles c d E, c F H seront semblabes, c'est pourquoy E c sera à c H comme d c à c F en composant E H sera à E c comme d F à d c & en raison inverse E c sera à E H comme d c à d F. par mesme raison à cause des paralelles E b, H I les triangles c E b, c H I seront semblables & en composant & renversant comme cy-devant E c sera à E H comme b c à b I, donc b c est à b I comme d c est à d F: mais comme d c est à d F de position ainsi b c est à b F de position, b c sera donc à b F comme b c à b I & par consequent b F & b I seront égales: mais au triangle F H I, b G est paralelle à la base H I & la ligne b G divise en 2 également la ligne F I au point b: elle divisera donc aussi en 2 également la ligne F H au point G, ce qu'il falloit prouver.

In the same situation as above: take the straight line FH parallel parallel to one of the extremes EA or ED of the 4 lines taken from the point E through the points of division of the line AD. I say that the line FGH is cut in two equal parts by the 3 other lines EA, EB, EC.
From the point F take the line Fd parallel to AD and from point H take HI parallel to EB—the middle one of the three lines—and with FH and Fd intersecting at I.
By the previous lemma the line Fd is cut harmonically into 3 parts by the points F, c, b, d: but because of the parallels Ed & FG the triangles cdE, cFG are similar, which is why Ec will be to cH as dc to cF and, adding them, EH will be to Ec as dF to dc, & in inverse ratio Ec will be to EH as dc to dF. Likewise, because of the parallels Eb, HI the triangles cEb, cHI will be similar; and adding and reversing as before Ec will be to EH as bc to bI. Hence bc is to bI as dc is to dF. But as dc is to dF so bc is to bF, so bc will be to bF as bc to bI and therefore b F and b I will be equal. In the triangle FHI, bG is parallel to the base HI and the line bG divides the line FI evenly in 2 at the point b: it therefore also will divide the line FH even in 2 at the point G. QED

I have no idea what the significance of the use of lower-case letters might be: none, I'd say. This proof is to do with what happens to 4 points in involution when one of them goes to infinity.

Lemme 4 (Fig 4)
Une ligne droitte B D étant couppée en 2 également au point C; si l'on prend quelque point A hors de cette ligne mesme si elle étoit prolongée, & ayant mené les lignes A B, A C, A D prolongées vers les parties de B D, si l'on tire par le point A la ligne I A H paralelle à B D: Je dis que la ligne droitte E H couppant les lignes A B, A C, A D, A H aux poins E, F, G, H sera couppée en 3 parties harminiquement en ces mesmes poins.
Que l'on mene par le point F la ligne droitte b F d, paralelle à B D qui sera divisée en 2 également en F: mais b F d & A H étant paralelles les triangles E b F, E A H seront semblables donc E F sera à E H comme b F à A H, mais comme b F est à A H ainsi F d qui est égale à b F sera à A H & à cause des paralelles F d & A H les triangles G F d, G H A seront semblables & par consequent comme F d est à A H ainsi G F est à G H, mais aussi comme F d est à A H ainsi E F est à E H donc E F est à E H comme G F est à G H, ce qu'il falloit démontrer.

The straight line BD is cut evenly in 2 at the point C. Take a point point A not lying on this line (even if it is extended), and have the lines AB, AC, AD extended from the parts of BD. Draw through the point A the line IAH parallel to BD. I say that the straight line EG cutting the lines AB, AC, AD, AH at the points E, F, G & H will be cut harmonically into 3 parts at these points.
Through the point F draw the straight line bFd, parallel to BD which will be divided evenly in two at F. bFd and AH being parallel, the triangles EbF, EAH will be similar, and so EF will be to EH as bF to AH. bF is to AH as Fd—which is equal to bF—will be to AH. Because of the parallels Fd and AH the triangles GFd, GHA will be similar, and so as Fd is to AH so GF is to GH. But also as Fd is to AH so EF is to EH. Therefore EF is to EH as GD is to GH. QED

I haven't got this ainsi/aussi thing at all sussed.

Corrolaire
De cecy il est évident que les lignes A I, A B, A C, A D, A H sont diposées de telle façon que de quelque maniere qu'on les couppe soit avec la ligne E H ou avec la ligne e I elles feront toûm;jours sur la ligne couppante 3 parties E F, F G, G H ou bien e F, F g, g I en sorte queces lignes seront ainsi couppées en ces trois parties harmoniquement pourveu que la ligne couppante couppe quatre de ces lignes: car si elle n'en couppoit que trois & qu'elle fut paralelle à une quatriéme elle seroit divisée par ces trois lignes en 2 parties égales par le Lemme troisiéme.

In this it is evident that the lines AI, AB, AC, AD, AH are are arranged in this way so that in some manner one cuts them with the line EH or with the line eI they will always cut the line in three parts—EF, FG, GH or eF, Fg, gI—such that these lines will be cut harmonically into 3 parts whenever the cutting line cuts 4 of these lines: but it if only cuts three and is parallel to the fourth it will be divided by the tree lines evenly in 2, but the third lemma.

I've got something badly wrong here: there are too many words for what I've come up with, and the expression telle façon comes up again later, suggesting that it's a piece of stock terminology.

Lemme 5 (Fig 5)
Une ligne droite C F estant couppée aux poins C, D, E, F, en trois parties harmoniquement: si l'on prend quelque point A hors de cette ligne, mesme si elle estoit prolongée & si ayant tiré des lignes A C, A D, A E, A F prolongées par le point A & par les poins de division de la ligne C D E F, on tire quelque ligne G M qui couppe ces 4 lignes aux poins G, H, L, M: Je dis que la ligne G M est couppée en 3 parties par les poins G, H, L, M harmoniquement.
Car ayant mené du point C la ligne C O paralelle à A F la ligne droitte C O sera couppée en deux également au point N par la ligne A D par le 3^me Lemme & par le Corrolair du 4^me les lignes A C, A D N, A E O, A F seront disposées de telle façon que la ligne droitte G M les couppant toutes quatre aux poins G, H, L, M elle sera divisée par ces mesmes poins en 3 parties harmoniquement, ce qu'il falloit démontrer.

A straight line CF is cut at points C, D, E, F harmonically into 3 parts. Take some point A not lying on this line (even if it is extended) and have the lines AC, AD, AE, AF extend from the point A through the points of division of the line CDEF. Then take some line GM which cuts these 4 lines at points G, H, L, M. I say that the line GM is cut harmonically into 3 parts by the points G, H, L, M.
By taking from the point C the line CO parallel to AF the straight line CO will be cut evenly into two at the point N on the line AD by the 3rd lemma and by the corollary of the 4th the lines AC, ADN, AEO, AF will be arranged in such a way that the straight line GM cutting all 4 of them at the points G, H, L, N will be divided harmonically into 3 parts by these points. QED

Scholie (Fig 6)
Mais si par les poins de division de la ligne C F on tire les lignes A C, A D, A E, A F toutes paralelles entr'elles: Je dis aussi que la ligne G M couppant ces 4 lignes aux poins G, H, L, M sera divisée par ces mesmes poins en 3 parties harmoniquement.
La demonstration ce cecy est claire: car à cause des paralelles les triangles I M F, I L E, I G C, I H D seront semblables & en composant & divisant leurs costez qui sont entr'eux en mesme proportion on fera comme la toutte F C à toutte M G ainsi la partie F E à la partie M L & comme D C à H G ainsi D E à H L donc M G à M L comme H G à H L car F C est donnée deposition à F E comme D C à D E.

Now through the points of division of the line CF draw the lines AC, AD, AE, AF. I still say that the line GM cutting these 4 lines at points G, H, L, M will be divided harmonically into 3 parts
The proof of this is clear: because of the parallels the triangles IMF, ILE, IGC, IHD will be similar, and adding and dividing their sides which are in the same proportion, one finds that as the whole FC to the whole MG so also the part FE to the part ML, and as DC to HG so DE to HL, therefore MG to ML as HE to HL, therefore MG to ML as HG to HL as FC is to be to FE as DC to DE.

OK, that's enough.

Misc

1

For future reference, should I ever need to name a demon: Anabibazon.

2

Also for future reference, should I ever find myself in the past trying to earn a living by anachronistically inventing logarithms: repeated multiplication by a base only fairly close to 1, together with linear interpolation, gives better results than I would have thought.

This:

#!/usr/bin/perl

my $BASE = 1.05;

sub logarithm
{
    my ($x) = @_;
    my ($x1, $x2) = (1, $BASE);
    my $y1 = 0;

    while ($x2 < $x)
    {
        ($x1, $x2) = ($x2, $x2 * $BASE);
        $y1 += 1;
    }

    return $y1 + (($x - $x1) / ($x2 - $x1));
}


print  "   x    |           log(x)           | %-age error\n";
print  "--------------------------------------------------\n";
foreach my $x (2 .. 10)
{
    my $logarithm = logarithm($x);
    my $log = log($x) / log($BASE);
    printf "%4.1f    |%10.3f    %10.3f    | %4.2f\n",
            $x, $logarithm, $log, 100 * abs($logarithm - $log) / $log;
}

Give this:

   x    |           log(x)           | %-age error
--------------------------------------------------
 2.0    |    14.203        14.207    | 0.03
 3.0    |    22.511        22.517    | 0.03
 4.0    |    28.407        28.413    | 0.02
 5.0    |    32.987        32.987    | 0.00
 6.0    |    36.719        36.724    | 0.01
 7.0    |    39.881        39.883    | 0.01
 8.0    |    42.614        42.620    | 0.01
 9.0    |    45.033        45.034    | 0.00
10.0    |    47.190        47.194    | 0.01

3

Regarding what I said about physics, it may be objected: “The bad end implausibly, the good end untidily. This is what theoretical physics means.”

4

I had in mind to write something about model theory, which I haven't done. But I did come across a footnote in A Thousand Plateaus that I'd not noticed before:

Historically, these have been the major problems of axiomatics: “undecidable” propositions (contradictory statements are also nondemonstrable); the powers of infinite sets, which by nature elude axiomatic treatment (“the continuum, for example, cannot be conceived axiomatically in its structural specificity since every axiomatization one can give it will rely on a denumerable model”). See Blanché, L'axiomatique, p. 80.

I assume the latter of these refers to Skolem's paradox: that although set theory covers uncountable sets, its axiomatization will admit countable models. This isn't strictly a paradox, in the sense of being a formal contradiction. It isn't necessarily even a problem. (I'm not sure what to make of the word ‘rely’ in the extract.) Hilbert's programme aimed at a more thorough paradox: grounding these infinities on finite objects (the formal proofs themselves).

But it is odd. Like the story of the man who went fishing with a picture of a worm, and caught a picture of a fish.

Speculative Realism s-r arc

Iain Hamilton Grant: nature philosophy

A claim:

The form of a thought is not independent of its content
The form ultimately derives from thought's content, its object: nature

This can be taken in one of two (contradictory?) directions:

The study forms of thought in themselves (e.g. linguistics, brain science, sociology of knowledge) could be regarded as a dubious enterprise insofar as they abstract away content
Universally valid forms of thought derive from the universe, so logic is physics

Suppose you are Goethe doing botanical studies. Through a long period of study of different flowering plants—and of particular plants in different states of development—through the use of the imagination to see plant forms related by a system of transformations—you allow the inherent logic of plant form to insinuate itself into your thinking. This gives you access to an idea of the plant that is objective in the sense of not belonging to you but to the plant itself. Appearing in thought is simply something that the plant does. (As a sideline perhaps, rather than as its day job.)

I think that there's a tension between saying that all structures of thought come from nature, and saying that a certain thought can be characterised as being objective by virtue of it taking its structure from its object's nature. In the latter case either thought is initially unnatural and can fail to obtain objectivity; or thought is initially objective but is prone to being denatured.

Many years ago I had an interest in John Lilly, a one-time NIMH researcher who invented the sensory deprivation tank (and so inspired the film Altered States, the source of my interest). I think he makes a good counter-example to this notion of objectivity. He pursued his investigation of the mind through naïve empiricism: self experimentation and self observation. He would spend hours in his tank, and then write up the resultant intense hallucinations. He progressed on to LSD, and then Ketamine. (I think it was around this point that he came to believe that he was a robot scientist sent back from the 25th century to observe 20th century human life.)

I remember reading an interview with Lilly in, I don't know, Omni or something. He was going on about the difference between ‘insanity’ and ‘outsanity’: where the latter is the consensual everyday world, and the former is the stuff in your (or rather his) head that it is difficult to talk about because it's so crazy. The reason I say that this is a counter-example is that this outcome looks largely determined by his method of investigation: a psychedelic cartesianism leading to mad dualism. In other words his thought was subjective in the sense of its content coming from its form, and not vice versa.

Perhaps the attempt to evacuate thought of content led to a cognitive equivalent video feedback, whereby the slightest remaining wisps of worldliness get amplified and mutated so that they seem to have a life of their own despite ultimately have an external source.

Graham Harman: Object-oriented philosophy

My initial curiosity was due a coincidence of words: Harman's is an object-oriented philosophy deriving from Heidegger's tool analysis. I thought I'd got the present-at-hand / ready-to-hand distinction through my thick skull. And a while back I was trying to think through the idea of Object Oriented Programming being Artificial Intelligence in drag, with both being based on a view of thought as the manipulation of representations of present at hand objects, leading to a suggestion that Rodney Brooks work in robotics and the emergence of agile software development (e.g. Extreme Programming) were parallel reactions. (I never got to the bottom of it.) I was worried that Harman might scramble what little I'd made of that all.

But, no, it wasn't anything like that.

There's the definition of an object: something which can be spoken of but that is not exhausted by what is said. (I think that was it. It now becomes clear why other people were taking notes.) It suggests that, e.g., if Boethius were right about music (that the theory is more perfect than the music itself) then music would fail to be an object. Come to think of it, if (big, up-front) object-oriented software design worked the way it was supposed to then objects would fail to be objects.

My mind's now gone rather blank, so I will make do with a couple of tangential remarks.

There was a claim that objects are distinguished by their qualities. There was an example given that, as real doubloons are different from imaginary doubloons, then they must differ in their qualities. There was a question as to how objects could be distinguished, since it was previously said that an object had an infinity of qualities. There was an answer that it was possible because there can be different sizes of infinity. This seemed needless and a bit random:

To judge the cardinality of a set of qualities implies that this set can be comprehended, which contradicts what was said about an object not being exhausted by what can be said about it.
To distinguish two objects only requires a single distinguishing quality to be found. It is identifying objects that surely becomes an infinite task, as it requires all the qualities. (cf Steven Vickers' geometric logic.)

And also there's the thing about index cards. Harman mentioned that is preparation he'd written the names of the four speakers on different cards, and formed different arrangements on his desk of groupings and contrasts. Also:

We now have five kinds of object […] five different types of relation […] three adjectives for what unfolds inside an object […] and three different kinds of noise […] a good initial model whose very strictness will smoke out those elements it might have overlooked.

I'm just noticing; it might be nothing. Ward Cunningham; Raymond Lull. That's all I'm saying.

robin2

Berlin vs. Göttingen

Screwdrivers

Probability and the infinite worlds

Holiday Reading

Biology

Tangency

TMA 2.2

The influence of science and technology on mathematical development in the 16th and 17th centuries

Thoughts

TMA 2.1

John Dee on navigation

Thoughts

Quines

TMA 1.2

Cube duplication in Greek mathematics

Footnotes

Additional References

Thoughts

TMA 1.1

(untitled)

Thoughts

MA290

Conics

Misc

1

2

3

4

Speculative Realism s-r arc

Iain Hamilton Grant: nature philosophy

Graham Harman: Object-oriented philosophy

The influence of science and technology on mathematical development in the 16^th and 17^th centuries