Who was the first philosopher to describe what we now call curve fitting or approximation?

Pierre Duhem discusses this a bit in Aim & Structure of Physical Theory, pt. 2, ch. 3 "Mathematical Deduction & Physical Theory", pp. 132-43, § 4 of which is the "Mathematics of Approximation":

a mathematical deduction is of no use to the physicist so long as it is limited to asserting that a given rigorously true proposition has for its consequence the rigorous accuracy of some such other proposition. To be useful to the physicist, it must still be proved that the second proposition remains approximately exact when the first is only approximately true.

What does it mean to be "approximately true"? Isn't something either true or false?

this “mathematics of approximation” is not a simpler and cruder form of mathematics. On the contrary, it is a more thorough and more refined form of mathematics, requiring the solution of problems at times enormously difficult, sometimes even transcending the methods at the disposal of algebra today.

This describes what machine learning (=sophisticated curve-fitting) is.

Certainly philosophers before Duhem (✝1916) have discussed this, but who was the first?

Cf. "Duhem's Bull"
See my related question: "Are mathematical suppositions of physical theories determined uniquely according to Aristotle and Plato?"
another very related question: "If nature is inherently imprecise, how is it so easy for us to conceptualize mathematical certainties?"

  • 2
    Approximate truth is a popular notion among realists, see SEP. It is not restricted to curve-fitting and the like and occurs already in Boole's Laws of Thought (1854):"Grant that the procedure thus established can only conduct us to probable or to approximate results; it only follows, that the larger number of the generalizations of physical science possess but a probable or approximate truth." He was not the first either.
    – Conifold
    Commented Jun 14 at 21:29
  • 2
    Newton in The Method of Fluxions (1671, published 1736) already writes "as in common Arithmetick we approximate continually to the truth, by admitting Decimal Parts in infinitum..."
    – Conifold
    Commented Jun 14 at 21:45
  • Mathematics is not true, it is useful. As such, there are simply more useful and less useful formulations. Get used to it :-) Alternative truths is another matter, not useful at all.
    – Scott Rowe
    Commented Jun 14 at 23:04
  • 1
    @ScottRowe If the math and reality correspond, that's truth. There are physically useful and physically useless mathematical deductions (cf. §§ 2 & 3 of Aim & Structure pp. 135-8).
    – Geremia
    Commented Jun 15 at 0:08
  • Certainly, with non-Euclidean geometry etc., it might not be able to be said that that sort of math "is concerned with matter in which perfect certitude is found" (In I Eth. l. 3 n. 36); cf. Kline's provocatively titled book Mathematics, The Loss of Certainty.
    – Geremia
    Commented Jun 15 at 0:21

1 Answer 1


Commenting on Aristotle, Metaphysics Chapter 3: 995a10, in the context of "The Method to Be Followed in the Search for Truth", St. Thomas Aquinas says:

  1. He [Aristotle] shows that the method which is absolutely the best should not be demanded in all the sciences. He says that the “exactness” [acribologia], i.e., the careful and certain demonstrations, found in mathematics should not be demanded in the case of all things of which we have science, but only in the case of those things which have no matter; for things that have matter are subject to motion and change, and therefore in their case complete certitude cannot be had. For in the case of these things we do not look for what exists always and of necessity, but only for what exists in the majority of cases [ut in pluribus].

Thus, probability/statistics is required for studying changing things.

  • Computing systems have no matter, but expecting perfection of them is apparently a forlorn hope. Also, as best I know, curve fitting is used in cryptography. It is the basis for a known secret that is hard for someone else to discover. Anti-knowledge?
    – Scott Rowe
    Commented Jun 15 at 1:00
  • @ScottRowe "Computing systems have no matter" Sure they do. All information has a material substrate.
    – Geremia
    Commented Jun 15 at 2:23
  • So mathematics is information and has matter. What is something that has no matter? We would have no information about it. What would we talk about then?
    – Scott Rowe
    Commented Jun 15 at 3:52
  • @ScottRowe "What is something that has no matter?" Being qua being, what metaphysics studies (the 3rd of the 3 degrees of abstraction; cf. end of this answer for details).
    – Geremia
    Commented Jun 15 at 3:57
  • But then it has no information, as you said, so we can never know about it. Your contention.
    – Scott Rowe
    Commented Jun 15 at 4:04

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