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In mathematics, first principles are referred to as axioms or postulates. Gödel's incompleteness theorems have been taken to prove, among other things, that no system of axioms that describe the set of natural numbers can prove its own validity - nor perhaps can it prove every truth about the natural numbers.
In philosophy, a first principle is a basic, foundational proposition or assumption that cannot be deduced from any other proposition or assumption.
First principles in formal logic
In a formal logical system, that is, a set of propositions that are consistent with one another, it is probable that some of the statements can be deduced from one another. For example, in the syllogism, "All men are mortal; Socrates is a man; Socrates is mortal" the last claim can be deduced from the first two.
A first principle is one that cannot be deduced from any other. The classic example is that of Euclid's (see Euclid's Elements) geometry; its hundreds of propositions can be deduced from a set of definitions, postulates, and common notions: all three of which constitute first principles.
Terence Irwin writes:
When Aristotle explains in general terms what he tries to do in his philosophical works, he says he is looking for "first principles" (or "origins"; archai):
- In every systematic inquiry (methodos) where there are first principles, or causes, or elements, knowledge and science result from acquiring knowledge of these; for we think we know something just in case we acquire knowledge of the primary causes, the primary first principles, all the way to the elements. It is clear, then, that in the science of nature as elsewhere, we should try first to determine questions about the first principles. The naturally proper direction of our road is from things better known and clearer to us, to things that are clearer and better known by nature; for the things known to us are not the same as the things known unconditionally (haplôs). Hence it is necessary for us to progress, following this procedure, from the things that are less clear by nature, but clearer to us, towards things that are clearer and better known by nature. (Phys. 184a10–21)
The connexion between knowledge and first principles is not axiomatic as expressed in Aristotle's account of a first principle (in one sense) as "the first basis from which a thing is known" (Met. 1013a14–15). The search for first principles is not peculiar to philosophy; philosophy shares this aim with biological, meteorological, and historical inquiries, among others. But Aristotle's references to first principles in this opening passage of the Physics and at the start of other philosophical inquiries imply that it is a primary task of philosophy.
Profoundly influenced by Euclid, Descartes was a rationalist who invented the foundationalist system of philosophy. He used the method of doubt, now called Cartesian doubt, to systematically doubt everything he could possibly doubt, until he was left with what he saw as purely indubitable truths. Using these self-evident propositions as his axioms, or foundations, he went on to deduce his entire body of knowledge from them. The foundations are also called a priori truths. His most famous proposition is I think, therefore I am, or Cogito ergo sum.
In physics, a calculation is said to be from first principles, or ab initio, if it starts directly at the level of established laws of physics and does not make assumptions such as empirical model and fitting parameters.