“Every definition implies an axiom, since it asserts the existence of the object defined.”

— Henri Poincaré, book Science and Method

Part II. Ch. 2 : Mathematical Definitions and Education, p. 131
Science and Method (1908)
Context: Every definition implies an axiom, since it asserts the existence of the object defined. The definition then will not be justified, from the purely logical point of view, until we have proved that it involves no contradiction either in its terms or with the truths previously admitted.

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Do you have more details about the quote "Every definition implies an axiom, since it asserts the existence of the object defined." by Henri Poincaré?

Henri Poincaré49

French mathematician, physicist, engineer, and philosopher … 1854–1912

Tom R. Burns (1937) American sociologist

Source: Systems theories (2006), p. 1.

“If we wish to express our ideas in terms of the concepts synthetic and analytic, we would have to point out that these concepts are applicable only to sentences that can be either true of false, and not to definitions. The mathematical axioms are therefore neither synthetic nor analytic, but definitions. …Hence the question of whether axioms are a priori becomes pointless since they are arbitrary.”

Hans Reichenbach (1891–1953) American philosopher

The Philosophy of Space and Time (1928, tr. 1957)

“A physical theory, like an abstract science, consists of definitions and axioms as first principles, and of propositions, their consequences; but with these differences:—first, That in an abstract science, a definition assigns a name to a class of notions derived originally from observation, but not necessarily corresponding to any existing objects of real phenomena, and an axiom states a mutual relation amongst such notions, or the names denoting them; while in a physical science, a definition states properties common to a class of existing objects, or real phenomena, and a physical axiom states a general law as to the relations of phenomena; and, secondly,—That in an abstract science, the propositions first discovered are the most simple; whilst in a physical theory, the propositions first discovered are in general numerous and complex, being formal laws, the immediate results of observation and experiment, from which the definitions and axioms are subsequently arrived at by a process of reasoning differing from that whereby one proposition is deduced from another in an abstract science, partly in being more complex and difficult, and partly in being to a certain extent tentative, that is to say, involving the trial of conjectural principles, and their acceptance or rejection according as their consequences are found to agree or disagree with the formal laws deduced immediately from observation and experiment.”

William John Macquorn Rankine (1820–1872) civil engineer

Source: "Outlines of the Science of Energetics," (1855), p. 121; Second paragraph

“Further, as every law of nature implies the existence of an invariant, it follows that every law of nature is a constraint.”

W. Ross Ashby (1903–1972) British psychiatrist

Source: An Introduction to Cybernetics (1956), Part 2: Variety, p. 130

“Definitions belong to the definers, not the defined.”

Toni Morrison book Beloved

Source: Beloved

“A: "Your objection to the self-evident has no validity. There is no such thing as disagreement. People agree about everything."
B: "That’s absurd; people disagree constantly, and about all kinds of things."
A: "How can they? There’s nothing to disagree about; no subject matter. After all, nothing exists."
B: "Nonsense. All kinds of things exist, you know that as well as I do."
A: "That’s one. You must accept the existence axiom, even to utter the term “disagreement.” But to continue, I still maintain that disagreement is unreal. How can people disagree when they are unconscious beings who are unable to hold any ideas at all?"
B: "Of course people hold ideas. They are conscious beings. You know that."
A: "There’s another axiom, but even so, why is disagreement about axioms a problem? Why should it suggest that one or more of the parties is mistaken? Perhaps all of the people who disagree about the very same point are equally, objectively right."
B: "That’s impossible. If two ideas contradict each other, they can’t both be right. Contradictions can’t exist in reality. After all, A is A."
Existence, consciousness, identity are presupposed by every statement and by every concept, including that of "disagreement." … In the act of voicing his objection, therefore, the objector has conceded the case. In any act of challenging or denying the three axioms, a man reaffirms them, no matter what the particular content of this challenge. The axioms are invulnerable.
The opponents of these axioms pose as defenders of truth, but it is only a pose. Their attack on the self-evident amounts to the charge. "Your belief in an idea doesn't necessarily make it true; you must prove it, because facts are what they are independent of your beliefs." Every element of this charge relies on the very axioms that these people are questioning and supposedly setting aside.”

Leonard Peikoff (1933) Canadian-American philosopher

Objectivism: The Philosophy of Ayn Rand (1991) ; Dialogue used to show that existence, conciousness, identity, and non-contradiction are axioms, using A as a defender of the axioms, and B as an opponent of the axioms,
1990s

“The core of a root definition of a system will be a transformation process (T), the means by which defined inputs are transformed into defined outputs. The transformation will include the direct object of the main activity verbs subsequently required to describe the system.”

Peter Checkland (1930) British management scientist

Source: Systems Thinking, Systems Practice, 1981, p. 223 as cited in: Gillian Ragsdell, Daune West, Jennifer Wilby (2002) Systems Theory and Practice in the Knowledge Age. p. 82. In the original quote Checkland summarised his earlier work with Smyth published in 1976.

“Whatever is referred to must exist. Let us call this the axiom of existence.”

John Rogers Searle Speech Acts: An Essay in the Philosophy of Language

Source: Speech Acts: An Essay in the Philosophy of Language (1969), P. 77.

“…the mathematician uses an indirect definition of congruence, making use of the fact that the axiom of parallels together with an additional condition can replace the definition of congruence.”

Hans Reichenbach (1891–1953) American philosopher

The Philosophy of Space and Time (1928, tr. 1957)

“Necessity may be defined in two ways, conformably to the two definitions of cause, of which it makes an essential part. It consists either in the constant conjunction of like objects, or in the inference of the understating from one object to another.”

David Hume book An Enquiry Concerning Human Understanding

§ 8.27
An Enquiry Concerning Human Understanding (1748)

“Every definition implies an axiom, since it asserts the existence of the object defined.”

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Henri Poincaré49

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