Poincaré’s Philosophy of Mathematics
After providing a short biographical sketch, we will presentPoincaré’s views organized as follows:
Philosophical Dictionary: PhiliaPoincare
(AlBiruni's contemporary Avicenna was not particularly a mathematicianbut deserves mention as an advancing scientist, as does Avicenna's discipleAbu'lBarakat alBaghdada, who lived about a century later.)AlBiruni has left us what seems to be the oldest surviving mentionof the Broken Chord Theorem (if M is the midpoint of circular arc ABMC,and T the midpoint of "broken chord" ABC, then MT is perpendicular to BC).
The great skill demonstrated by Ptolemy and his predecessors indeveloping their complex geocentric cosmologymay have science since in factthe Earth rotates around the Sun.
Science and Hypothesis: The Complete Text: Henri …
When a law has received a sufficient confirmation from experiment, wemay adopt two attitudes: Either we may leave this law in the fray; itwill then remain subject to incessant revision, which without anydoubt will end by demonstrating that it is only approximate. Or elsewe may elevate it into a principle by adopting conventions.(Poincaré 1905b: 165–166; 1913b: 335)
Poincaré distinguished empirical laws from conventionalprinciples. How does one move from “simple” empiricallaws, understood as confirmable hypotheses, to principles asthe result of a conventional decision to withdraw a confirmablehypothesis from the judgment of experience? In mechanics he employsthe same methodological procedure as in geometry with regard toclasses of displacements and groups:
Science and Hypothesis : Henri Poincaré  Internet Archive
An answer to this difficulty can be found by examining four steps ofPoincaré’s psychophysiological reconstruction of thegenesis of geometry. In his early articles, Poincaré arguesthat geometry concerns only the relations expressed in the axioms, notsome inherent features of the primitives:
What we call geometry is nothing but the study of formal properties ofa certain continuous group; so we may say, space is a group.(Poincaré 1898: 41)
LibriVox recording of Science and Hypothesis, by Henri Poincaré

Science and Hypothesis, by Henri Poincaré : chapter3
This philosophy of science provided a significant impetus for, but Poincaré himself criticized the in () (1912).

[Henri Poincaré] ⋗ Science and Hypothesis ⋮ Books Online
Poincaré argues that

Science and Hypothesis by Henri Poincaré  Read Online
As Philippe Nabonnand has remarked, Poincaré’spresentation of geometrical space is as a whole in fact circular:
Science and Hypothesis by Henri Poincaré
AlKindi (called Alkindus or Achindus in the West)wrote on diverse philosophical subjects, physics, optics,astronomy, music, psychology, medicine, chemistry, and more.
Genjiro said: This book by Poincare was mostly over my head
AlHassan ibn alHaytham (Alhazen)made contributions to math, optics, and astronomywhich eventually influenced Roger Bacon, Regiomontanus, da Vinci, Copernicus,Kepler, Galileo, Huygens, Descartes and Wallis,thus affecting Europe's Scientific Revolution.
Henri Poincaré (Stanford Encyclopedia of Philosophy)
Eudoxus has been quoted as saying"Willingly would I burn to death like Phaeton, were thisthe price for reaching the sun and learning its shape,its size and its substance."
Aristotle is considered the greatest scientist of theancient world, and the most influential philosopher and logician ever;he ranks #13 on Michael Hart's list of the Most Influential Persons in History.
Poincare opens a path lying between a realist ..
His PoincaréBirkhoff Fixed Point Theorem is especially importantin celestial mechanics, and led to instant worldwide fame:the great Poincaré had described it as most important,but had been unable to complete the proof.
Science and Hypothesis by Henri Poincar
He wrote prodigiously on all scientific topics (his writings are estimatedto total 13,000 folios); he was especially noted forhis comprehensive encyclopedia about India, and ,which starts from notions about shadows but develops much astronomyand mathematics.
Science and Hypothesis : Jules Henri Poincare : …
Since his famous theorems of geometry were probably already knownin ancient Babylon, his importance derives from imparting thenotions of mathematical proof and the scientific methodto ancient Greeks.