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.

  • $\begingroup$ But must all paths converge on Rome (i.e. modern mathematics)? $\endgroup$ Jul 14 at 16:23
  • $\begingroup$ 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. $\endgroup$
    – NWR
    Jul 14 at 16:41
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    $\begingroup$ 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. $\endgroup$
    – Conifold
    Jul 14 at 20:10
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    $\begingroup$ Teleological fallacy? $\endgroup$ Jul 16 at 6:04
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    $\begingroup$ See Whig Interpretation of History $\endgroup$ Jul 16 at 14:30

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.

  • $\begingroup$ 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. $\endgroup$ Jul 18 at 16:11
  • $\begingroup$ @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? $\endgroup$ Jul 18 at 19:53
  • $\begingroup$ 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. $\endgroup$ Jul 19 at 2:15
  • $\begingroup$ 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. $\endgroup$ Jul 19 at 2:23
  • $\begingroup$ @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. $\endgroup$ Jul 19 at 2:27

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