I would say the same of the conditions imposed by symmetry. All these hypotheses form, so to speak, the foundation common to all the theories of mathematical physics. Finally, apparent hypotheses are definitions or conventions rather than actual claims about the world. They therefore may not be considered to be hypotheses at all, although they are, however, often mistaken for hypotheses. Although he presents arithmetic in Science and Hypothesis without mentioning hypotheses explicitly, we can see a role for them in his analysis here as well, which we will present in the order that he follows.

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The principle of mathematical induction is necessary to prove general statements about numbers and is thus a necessary tool in mathematics. Therefore the principle of mathematical induction is similar to a natural hypothesis or at least plays a similar role to that of a natural hypothesis.

Of course, there are important differences between mathematical induction and the principle of induction used in the natural sciences. The principle of mathematical induction gives certainty, while the inductive generalizations in the natural sciences give only probability 27; 3. The existence of consistent alternative geometries shows that the geometry of space cannot be determined a priori. Thus, statements about the geometry of space are apparent hypotheses or conventions that appear to be making descriptive claims, however they are actually closer to definitions, though they cannot be reduced to being merely linguistic.

Thus sensible space can be seen as an example of a natural hypothesis. We introduce a metric to this space in order to measure distance by using the language of group theory, that is, the mathematical study of abstract algebraic structures. Groups serve as a tool of reasoning about representations of muscular sensations.

We can choose a group or sub-group which corresponds to the geometry of Euclid, Lobachevskii or Riemann to describe space. The empirical underdetermination of the groups has a purely contrastive arbitrary character if the groups are isolated from the background beliefs furnished by our mind Stanford In contrast to geometry, no choices between alternatives are given in arithmetic. We have thus far discussed arithmetic and geometry and examined a priori principles, natural hypotheses, and apparent hypotheses. What then is a good experiment?

It is one that informs us of something besides an isolated fact, one that allows us to make predictions; that is, that allows us to generalize. For without generalization, we cannot make predictions. The hypotheses that arise from experiment are verifiable hypotheses. The process of generalization presupposes a belief in the unity and simplicity of nature see ff.

Any good experiment in physics can be generalized in an infinite number of ways, depending on our preconceived ideas, and every generalization results in a hypothesis. Not all laws are directly accessible to experiment. Generalization in physical science has to take the mathematical form of a differential equation in order to grasp complexity and to unify science:.

It is not sufficient for each elementary phenomenon to obey simple laws, all those to be combined must obey the same law as well. The law of inertia, for example, obtains only with respect to the pre-existing category of differential equations.

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These differential equations are, in the final analysis, the physical laws:. Newton has shown that a law is only a necessary relation between the present state of the world and its immediately subsequent state. All the laws discovered since are nothing else; they are in sum, differential equations.

### Henri Poincaré

In mechanics, the categorical tools are the principle of empirical induction and differential equations. The scheme of generalization in mechanics appears as follows: Experience provides complex phenomena, i. From these elementary phenomena, we generalize e. These conventions do not constitute a choice between physical alternatives, but rather must be interpreted as defining the basic terms of a theory.

VI and VII. In the physical sciences proper, such as optics and electrodynamics, the scene changes entirely and principles seem no longer to share the conventional character of the geometrical postulates.

Indifferent hypotheses differ from geometrical conventions in one very important sense. While geometry rests on a conventional decree, the structures of physics mechanics included do not systematically rest on indifferent hypotheses; rather, these hypotheses have only a psychological and pedagogic function as mental constructs that are in a sense superfluous, since they can change while leaving the experimental results intact.

Here then is the methodological analogy and link between the principle of induction, conventions in the sense of apparent hypotheses and verifiable hypotheses.

## Henri Poincaré

All are guided by experience without being entirely experimental, and all involve presupposed categories or natural hypotheses. Verifiable hypotheses employ conventional elements in the generalization process and may presuppose the empirical principle of induction. Geometrical conventions are apparent hypotheses guided by the experience of muscular sensations and involve the category of groups.

The indifferent hypotheses are conventional in the ordinary sense stipulations of ontological entities—mechanical models that increase our theoretical insight, while the physical principles are either verifiable hypotheses elevated by decree beyond the crucible of experiment or well-founded with respect to the network of actual science.

Historically, both theses are directed broadly against the founders of modern logic and set theory such as Cantor, Peano, Frege, Russell, Zermelo, and Hilbert. On closer inspection, the historical situation is more complicated. With respect to Georg Cantor, for example, we must distinguish his work as a logical-set theoretical method useful in treating mathematical problems especially in the theory of functions on the one hand, and on the other hand, set theory as a tool for foundations or for considerations on infinite cardinality.

## Henri Poincaré

There is no actual infinity, the Cantorians have forgotten that, and they have fallen into contradiction. It is true that Cantorism rendered services, but that was when it was applied to a real problem whose terms were clearly defined, and we could walk safely. Logisticians as Cantorians have forgotten. It would be false, however, to believe that he thus conflates logic and psychology.

## Books by Poincaré, Henri (sorted by popularity) - Project Gutenberg

As we have already seen, in arithmetic pure intuition is necessary to justify the complete or mathematical induction principle. In The Value of Science , he distinguished three kinds of intuition: an appeal to sense and to imagination, generalization by induction, and intuition of pure number—whence comes the axiom of induction in mathematics. He was evidently right given that not only is the modern logic of predicates richer than the traditional syllogistic logic, but in order to carry out the logicist project, one is even led to expand logic again by certain set-theoretic postulates of existence. According to this view, a gap in a proof is no longer a logical gap, but a gap in the mathematical understanding Detlefsen , Russell []: Russell succeeded in developing a theory that respects his principle, the ramified type theory.

He did not seriously take up the question and only formulated successively different variants of the principle when he learned from Ernst Zermelo that the proof, proposed by Argand and Cauchy, of the fundamental theorem of algebra makes an appeal precisely to the rejected, non-predicative definitions. The absence of a truly convincing solution shows that the analysis of the concept of predicativity with the help of informal methods appeared to have reached its limits. According to the historians, the modernists are thos eat the beginning of the 20th century who supported an algebraization of the contents up to and including the language of signs, while the conservative tendency finds the justification of mathematical knowledge before language, that is, in the intellect.

Since the death of Bourbaki, it is fashionable to emphasize the intuitive aspects of mathematics. For him, mathematical reasoning possesses and must conserve an intuitive content that defies the formal contents. Rather, he changes the terms of the Kantian contrast, emphasizing the balance between exactness, that which is common to many minds and is transmissible by discourse, and subjective acquaintance, which is intransmissible sensation. He expresses this difficult balance in a formula well known due to its popularization by Einstein:.

It is by distancing themselves from reality that they acquired this perfect purity. In other words, the state of the bodies and their mutual distances at a given instant will depend only on the state of these same bodies and their mutual distances at the initial instant, but will in no way depend on the initial absolute position and initial absolute orientation of the system.

Both attempt to account philosophically for the incompleteness of scientific objects.

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