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  • Science and Hypothesis by Henri Poincare
  • John Safranek
POINCARE, Henri. Science and Hypothesis. Translated by Melanie Frappier, Andrea Smith, and David J. Stump. New York: Bloomsbury Press, 2018. xxvii + 171 pp.

Bloomsbury Press has decided to reintroduce the English-speaking world to Henri Poincare's Science and Hypothesis. Poincare was a turn-of-the-century mathematician who made significant contributions in the fields of theoretical physics and the philosophy of science.

This volume is a collection of fourteen essays published during Poincare's lifetime. Poincare upholds the relevance of hypotheses for both mathematics and physical science. Against rationalists, he argues that hypotheses are not first principles; against empiricists, he argues that hypotheses are not learned from experience.

Poincare asserts that experiment is the only source of truth because only it can offer something new and certain. This raises the question of the role of mathematical physics. Poincare claims that observation alone is insufficient because scientists must also generalize the results of their conclusions. It is not enough to say that this particular result occurred in these specified circumstances at this time. A scientist must take the results, analogized to a pile of stones, and organize them into a coherent form, such as forming the stones into a house.

The scientist must generalize so that he can predict, and prediction requires generalization. Such generalizations can err, but often high levels of probability are possible. Any generalization presupposes the unity and simplicity of nature. If one part of nature were distinct from all others, we would know only one part. And if a simple law has been observed in many particular cases, then we can suppose it applicable to analogous cases.

Generalizations are significant because they are hypotheses and thus are necessary for science: thus the title of this essay collection. Although each hypothesis must be subject to verification, lack of verifiability is not a setback for science, but rather an advance because more knowledge has been obtained. Poincare identifies three types of hypotheses: the natural, from which we cannot escape and are foundational for mathematical physics, for example, that the effect is a continuous function of the cause; the indifferent, that is, an initial assumption one is aware one is making or mechanical model we develop to explain observed phenomena: an analyst making calculations assumes at the outset that matter is either continuous or, conversely, comprised of atoms; and true generalizations, which are the focus of the experiment that must be confirmed or invalidated: these are the empirically verified laws.

Central to Poincare's thesis is that scientists use conventions that are neither a priori nor empirical. He argues that mathematics uses mathematical induction to make universal claims. In starting with a few basic mathematical definitions and using mathematical induction, we can assert claims about the infinite natural numbers. In so doing, we transcend our experience, thus generating a synthetic a priori judgment.

Poincare also discusses the development of mathematical physics. He claims that scientists always divide the complex phenomenon into many [End Page 398] elementary phenomena by decomposing it with respect to time and space. Regarding time, the scientist focuses only on the immediate past, disregarding more distant times. With space, the scientist focuses only on a very small region of space, such as focusing on heat exchange only between two contiguous points. This may not always be true, but when it is verified, it allows the scientist to do mathematical physics by making successive approximations. If the phenomenon does not stand the test of verification, then the scientist must search for something analogous to reach the elementary phenomenon, for example, an observed motion can be decomposed into simple ones, such as sound into its harmonics.

Mathematical physics are possible due to what Poincare terms "the superposition of a great number of particles that are all alike." This allows for the introduction of differential equations. The near homogeneity of matter, that each elementary phenomenon obeys the same simple law, allows mathematics to be applied to the phenomena.

Poincare's fourteen essays collected in Science and Hypothesis are divided into four main sections by the authors: part 1, Number and Magnitude, consists of essays "On the Nature of Mathematical...

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