Are the laws of arithmetic inductive truths?

After the considerations have been adduced, we make itprobable that numerical formulae can be derived from the definitions of theindividual numbers alone by means of a few general laws, now we must ascertainthe nature of the laws involved.

Mill calls the principle “The sums of equals areequals” an inductive truth and a law of nature of the highest order, and inorder to be able to call arithmetical truths laws of nature, mill attributes tothem a sense which they do not bear. He always confuses the applications thatcan be made of an arithmetical proposition, which often are physical and dopresuppose observed facts, with the pure mathematical proposition itself.

The general laws of addition cannot be laws of nature.The numbers, moreover, are related to one another quite differently from theway in which the individual specimens of, it is in their nature to be arrangedin a fixed, definite order of precedence. And each one is formed in its ownspecial way and has its own unique peculiarities.

Inductions must base itself on the theory ofprobability, since it can never render a proposition more than probable, todevelop the theory of probability, we must presuppose arithmetical laws. Byinductions we can only to encounter content, we can understand by induction amere process of habituation, in which case it has of course absolutely no powerwhatever of leading to the discovery of truth, whereas the numbers areliterally created, and determined in their whole natures, by the process ofcontinually increasing by one.

Leibniz holds the opposite view from Mill, the truthsof number are in us, they must have principles whose proof does not depend onexamples and therefore not on the evidence of the senses.

Are the laws of arithmetic synthetic a priori oranalytic?

The laws of arithmetic are not posteriori, they mustbe synthetic a priori or analytic. Kant Baumann and Lipschitz are declared forthe former, assert that propositions and number are derived from innerintuition. For Hankel, the numbers are intuition of magnitudes, but it is hardto allow an intuition of 100,000, it is necessary to examine the word intuition.For Kant, in the sense of the Logic, we might perhaps be able to call 100,000an intuition, but an intuition in this sense cannot serve as the ground of ourknowledge of the laws of arithmetic.

Arithmetic is not akin to geometry, in geometry, it isquite intelligible that general propositions should be derived from intuition, thepoints or lines or planes which we intuit are not really particular at all, whichis what enables them to stand as representatives of the whoe of their kink, butwith numbers it is different, each number has its own peculiarities.

Empirical propositions hold of what is physically orpsychologically actual, the truths of geometry govern all that is spatiallyintuitable, whether actual or product of our fancy, so long as they remainintuitable, still subject to the axioms of geometry. But conceptual thought isdifferent from empirical propositions, conceptual thought, like the basis ofarithmetic lies deeper, it seems, than that of any of the empirical sciences, andeven than that of geometry. The truths of arithmetic govern all that is numerable,and the laws of number are connected very intimately with the laws of thought.

For Leibniz, the laws of number are analytic. For W.S.Jevonsalgebra is a developed logic, and numberbut logical discrimination.

For mathematics, we can follow the logical rules to usesymbols to calculate, without knowing anything intuitable, or with which we couldbe sensibly acquainted. All we need to know is how to handle logically thecontent as made sensible in the symbols and, if we wish to apply our calculusto physics, how to effect the transition to the phenomena.

To use deduction, we can leave the fact where it is,while adopting its content in the form of a condition, by substituting in thisway conditions for facts throughout the whole of a train of reasoning, we shallfinally reduce it to a form in which a certain result is made dependent on acertain series of conditions. This would make the laws of number analyticjudgements.