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I have noticed a tendency among some historians and scholars of mathematics to regard the mathematics of antiquity as a less developed version of modern mathematics. This view reminds me of the belief that evolution is directional and was bound to produce the species we know today.

A corollary to this belief, in my opinion, is that if the body of knowledge we now call modern mathematics had not emerged, then the mathematics of antiquity would have stagnated.

Rodrigo de Azevedo
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Euclid Looked On Beauty Bare
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  • But must all paths converge on Rome (i.e. modern mathematics)? – Euclid Looked On Beauty Bare Jul 14 '21 at 16:23
  • I would think that those paths guided by algebra are often shared by those guided by geometry, but the relative progress down these paths made by these different approaches would be contingent on many historical factors unrelated to mathematics. I hope that doesn't sound too cryptic. – nwr Jul 14 '21 at 16:41
  • Similar to the (mistaken) belief that biological evolution strived to culminate on humans. Bacteria are just as evolved as we are. – vonbrand Jul 14 '21 at 17:57
  • Determinism. Al math is determined to evolve in one direction only That which the modern mathematicians contemplate. Its a form of realism also encountered in physics. Reality forms physical theories. Likewise, mathematical reality (Plato) directs mathematical theories. These theories can only approximate that reality. Naive realism would also be appropiate. I think its more a philosophical question. – Deschele Schilder Jul 14 '21 at 18:28
  • @DescheleSchilder If reality selects physical theories and physical theories are selected from mathematical theories, then mathematical theories are selected (indirectly) by reality. In other words what we can know of Plato's world depends on what we consider is real or meaningful or relevant in our lives. – Euclid Looked On Beauty Bare Jul 14 '21 at 19:13
  • Yes. Thats a way of looking at it. But I think this would rob physics of substance. Math refers to math objects only (in the ideal Platonian world). Physics refers to "hard" substance. The two worlds (of substance and idea) can be connected. And indeed, math connects them. – Deschele Schilder Jul 14 '21 at 19:31
  • @Nick I have read that some scholars of the 17th century such as Hobbes and Newton considered the new algebraic geometry of Descartes as only a supplement to the older descriptive geometry of Euclid. This stance has puzzled me. Were they just being old fashioned or sentimental or did they sense a part of mathematics, a part of Plato's world, was being sealed off with the uncritical application of algebra to geometry? – Euclid Looked On Beauty Bare Jul 14 '21 at 19:35
  • @DescheleSchilder a knower does the connecting as long as reality does not extinguish the knower. – Euclid Looked On Beauty Bare Jul 14 '21 at 19:52
  • or at least does not extinguish his or her capacity to know. – Euclid Looked On Beauty Bare Jul 14 '21 at 19:57
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    That the past is to be understood as a precursor to the present is generally called Whiggish historiography. Grattan-Guinness coined the term history as heritage for treating past concepts as imperfect prototypes of modern ones more specifically. Some other keywords are anachronism, march of progress and so on, see Current ways of thinking in the History of Mathematics for discussion. – Conifold Jul 14 '21 at 20:10
  • That goes without saying... It would be difficult to conceive reality without a conceiver. But even the conceiver can be deceived. The conception or inception can be deception. History could have developed in a different ways. – Deschele Schilder Jul 14 '21 at 20:22
  • Isnt the past always a precursor of the present? – Deschele Schilder Jul 14 '21 at 23:01
  • People who dont beliieve this say that if present day math hadnt been developed a different math would have evolved. There can be a different physics and math adhered to by alien cultures in other places in the universe. Who says ours is the only one? Nature itself? But then again, how looks Nature itself? Is there a unique independent reality? Of course there is. But it depends on our views. – Deschele Schilder Jul 14 '21 at 23:11
  • That sounds about right to me. Newton's absolute space was Euclidean and the methods of Descartes allowed him to present his physics geometrically rather than analytically. The Greek preference for geometry still resonating at Newton's time. It was a new language for doing Euclidean geometry. – nwr Jul 15 '21 at 21:45
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    Teleological fallacy? – Anton Sherwood Jul 16 '21 at 06:04
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    See Whig Interpretation of History – Mauro ALLEGRANZA Jul 16 '21 at 14:30
  • Teleological explanations should not be rejected categorically, but they should be stated explicitly so that they can be thoughtfully evaluated, criticized and tested. – Euclid Looked On Beauty Bare Jul 18 '21 at 15:37
  • Do you mind proving this "corollary" or providing a reference? – Moishe Kohan Jul 19 '21 at 21:06
  • I am using corollary in a casual manner. See this wikipedia article on various conceptions of corollary. So I suppose my corollary is more like a plausible conjecture which could be refuted. – Euclid Looked On Beauty Bare Jul 20 '21 at 12:36
  • @EuclidLookedOnBeautyBare: Then you should give at least one example of somebody holding the belief stated in the corollary, I cannot think of any. To me, there is no logical connection of any kind between the stated assumption and the "corollary." – Moishe Kohan Jul 20 '21 at 13:52
  • I hold the belief. I will rewrite the corollary to make this clear. – Euclid Looked On Beauty Bare Jul 20 '21 at 14:56
  • @MoisheKohan, an associated question is why did historians adopt a whiggish view of ancient mathematics? – Euclid Looked On Beauty Bare Jul 29 '21 at 16:09
  • I find your question to be too imprecise. With one interpretation, modern math contains all of the ancient (say, Greco-Roman) math, which makes the "Whiggish" viewpoint manifestly true. As for the "corollary", it is one of the "counter-factual" history questions which are not objectively answerable. For instance, suppose humanity were to be stuck in Dark Ages forever (might still happen!) with all the ancient knowledge eventually lost. This development cannot even be called a "stagnation," but regress. This makes your "corollary" (in my mind) manifestly false. – Moishe Kohan Jul 29 '21 at 16:18
  • With another interpretation of the main question, most of the modern math cannot be regarded as a "development" of the ancient math, as it represents complete break with it. For instance, topology, PDEs, Probability Theory, set theory, category theory... There is nothing in "ancient math" that can be regarded as a "less developed version of these areas," thereby making the main claim (of some unnamed historians) manifestly false. What do these historians have in mind, I do not know since your question contains no citations, no references. – Moishe Kohan Jul 29 '21 at 16:26
  • "With one interpretation, modern math contains all of the ancient (say, Greco-Roman) math, which makes the "Whiggish" viewpoint manifestly true." this is a truism of Whig history but it does not explain why whigs hold it to be true. My conjecture is that they hold it to be true because they implicitly believe in a counterfactual claim that ancient math would have stagnated. – Euclid Looked On Beauty Bare Jul 29 '21 at 16:28
  • Counterfactual claims are actually quite common because they help to make sense of the world. For example Sally made the right to decision to quit her job 5 years ago because she would be worse off today if she kept it. – Euclid Looked On Beauty Bare Jul 29 '21 at 17:15
  • A more scientific example concerns the history of climate change. The associated counterfactual claim of carbon dioxide induced climate change is that the climate of today would be less different from the climate of 150 years ago had we not burned fossil fuels. – Euclid Looked On Beauty Bare Jul 29 '21 at 17:39
  • I do not understand your questions, I have to say. Are you asking why some people believe that modern math contains all of the (extant) ancient math? Because it is an easily provable fact once you accept the axiomatic method. (With the usual caveat that, say, Euclid's treatment of geometry had some deficiencies which were rectified in the 19-20th century: Some of his arguments relied on pictures, while complete proofs required extra axioms that Euclid missed.) – Moishe Kohan Jul 29 '21 at 17:46
  • If I write a "modern" math paper dealing with, say, CAT(0)-spaces (whatever these are), I am free to quote any of Euclid's theorems I like. (I would be comparing, say, geometry of triangles in a CAT(0) spaces to the geometry of Euclidean triangles.) Sadly, not that many of the ancient theorems would be useful for the modern research, but they all remain available on need-to-use basis. On the other hand, if I were to write a paper on geometry of planar triangles in Euclid's style, it is highly unlikely that I will be able to publish it in any professional math journals. – Moishe Kohan Jul 29 '21 at 17:52
  • Regarding your counter-factual examples: Not every good question is suitable for Stack Exchange (including HSM). This is how SE was designed by Jeff Atwood and Joel Spolsky. – Moishe Kohan Jul 29 '21 at 17:56
  • Pasch's axiom is supposedly a missing axiom, but I don't think Euclid elements is in need of such an axiom as long as Euclid's original text is followed. It was 19th century reformulations of Euclid elements that gave rise to the need for Pasch's axiom. – Euclid Looked On Beauty Bare Jul 30 '21 at 15:42
  • "Euclid's treatment of geometry had some deficiencies which were rectified in the 19-20th century" The deficiencies arose in the minds of some influential 19th mathematicians because they had a whiggish view of Euclid's elements. Euclid's reliance on pictures is not a weakness. However, the value of pictures is undermined when the wrong pictures are used to represent ideas in the elements. Playfair's axiom as a picture is NOT equivalent to Euclid's picture of the 5th postulate when the latter picture is constructed in accordance with Euclid's definitions and postulates. – Euclid Looked On Beauty Bare Jul 31 '21 at 14:40
  • A well drawn picture is like a well drawn circuit diagram where no relevant element is left to the imagination. When Euclid says a straight line is drawn through two points, then a well drawn picture consists of a line which connects two marks or dots which symbolize those points. – Euclid Looked On Beauty Bare Jul 31 '21 at 14:55

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It's not that earlier mathematics is a less developed version of contemporary mathematics but that earlier mathematics is a precursor to modern mathematics. We cannot have later mathematics without earlier mathematics and it stands to reason that this earlier mathematics will be less developed.

Mathematics, because it is a subject thought about and debated about by human beings will demonstrate a telos because people have a telos. Hence it is not wrong for historians of mathematics to discern a certain thread or threads along which mathematics develop.

For example, since Riemann, the notion of a manifold took about fifty years to crystallise into definite form. And it's also ramified into many other directions: orbifolds, foliations, bundles, sheaves and toposes.

Mozibur Ullah
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You can call it whatever you like, depending on your attitude towards it. From a fallacy to an absolutely objective Platonic realism.

I think it is a naive realistic view on history. It is the same realism as expressed by many physicists. A reality is assumed which pulls our thoughts in the right direction. Now this can of course not be denied. It is the question though how we know that our theories are indeed rightly pulled. Just stating that this happens automatically in the course of time is, well, naive. History shows that this is not the case and ignoring history or even subject it to a suppose rule is, well, naive.

This view inhibits progress as new theories, mathematical or not, are excluded. Views that diverge are seen as not real and will consequently be put aside, ignored, or even laughed at and ridiculed.

So one can call the view a conservative anti-revolutionary Platonic realism, in favor of the existing modes of thinking.

Deschele Schilder
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  • Since the inception of algebraic geometry in the 1600s, the logic of algebra has been pulling the development of geometry in a certain direction. So in the time period when the logic of algebra came to dominate mathematics, the mathematics at the start of the algebraic revolution can be described as a precursor to later mathematical developments, but it is wrong to generalize this for all of history. – Euclid Looked On Beauty Bare Jul 18 '21 at 16:11
  • @EuclidLookedOnBeautyBare what is the connection between algebraic geometry and geometry? The former is about the solution of say m polynomials of say n variables in a certain field (say the reals) set to zero. The solutions of the variable tuples can be represented in an n-dimensional space, but does that say anything about the structure of that space, its geometry? – Deschele Schilder Jul 18 '21 at 19:53
  • The application of algebraic methods to Euclidean geometry produced a new kind of geometry, which we now call Cartesian geometry. Cartesian geometry gave rise to the so called non-Euclidean geometries, but it would be more accurate to call these Non-Cartesian geometries. – Euclid Looked On Beauty Bare Jul 19 '21 at 02:15
  • In my opinion the term geometry should be reserved for Euclidean geometry or perhaps Hilbert's and Tarski's axiomatic, definition free, geometries. The other geometries should be called manifolds. – Euclid Looked On Beauty Bare Jul 19 '21 at 02:23
  • @EuclidLookedOnBeautyBare There is also non-Euclidean geometry. That is in fact the geometry of the space(time) around us. You can call that a manifold of course. – Deschele Schilder Jul 19 '21 at 02:27
  • @EuclidLookedOnBeautyBare But what is the basic difference beetween Euclidean space and non-Euclidean space? Except that the distances can vary in the non-Euclidean space? So parallel lines will converge and pi can vary? All Euclidean stuff (axioms) can be defined in the non-Eucliden space also. Which makes it Euclidean again. – Deschele Schilder Jul 19 '21 at 03:33
  • The idea that a straight line is the "shortest distance between two points" doesn't appear anywhere in Euclid's Elements but it appears in Cartesian geometry and so called "non-Euclidean" geometries. – Euclid Looked On Beauty Bare Jul 19 '21 at 16:16
  • @EuclidLookedOnBeautyBare But what about the 3d Euclidean and non-Euclidean space? In both, the idea of a straight line as the shortest distance does not appear in Euclid (had he known of the curved space). But do you think that he would both see as the same? – Deschele Schilder Jul 19 '21 at 16:31
  • No, I don't think he would. Also, if something is the same as the original, why give them different names? – Euclid Looked On Beauty Bare Jul 19 '21 at 16:44
  • This is why I posed the question " what did Euclid mean by a straight line in his time?" At the present time only three approaches seem to be allowed:
    1. A straight line is the "shortest distance...." which Euclid did not say.
    2. In the axiomatic tradition the content of definitions is considered irrelevant so the definition of a straight line is considered irrelevant.
    3. Argue that the original version of Elements by Euclid was strictly axiomatic and never contained any definitions so Euclid never wrote a definition of straight line.
     – Euclid Looked On Beauty Bare Jul 19 '21 at 16:51
  • @EuclidLookedOnBeautyBare If it is the same then there is no use. If they are different yes. You can built a house with straight bricks in a straight space and in a curved space. In the straight space (Euclidean) there will be no tension inside the bricks. If you use them in a curved space there will be tension if you use them. – Deschele Schilder Jul 19 '21 at 16:59
  • @EuclidLookedOnBeautyBare While typing on my phone you typed too. Or somewhat sooner. :) But what was E's attitude towards a straight line? I now its why you asked exactly that but what do you think now? Housebuilding? – Deschele Schilder Jul 19 '21 at 17:10
  • @DesheleSchilder, sorry, I clicked "automatically move this discussion to chat" to see what would happen and it automatically suspended you which I did not intend. – Euclid Looked On Beauty Bare Jul 20 '21 at 13:14
  • @EuclidLookedOnBeautyBare I cant take part in chat. Im suspended on the physics site. I asked the wrong questions there. Im not sure what that has to docwith this site though bu beïng suspended on one site apparantly has that effect.... – Deschele Schilder Jul 20 '21 at 13:31