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“We have several other problems and theorems, and several consequences deducible from the preceding; but the mistrust which I have of my slight experience and capacity does not permit me to advance more till my present effort has passed the examination of able men who may oblige me by looking at it.
It is a ratiocination; and when our knowledge of the causes is so perfect, as to extend to the exact numerical laws which they observe in producing their effects, the ratiocination may reckon among its premisses the theorems of the science of number, in the whole immense extent of that science.
Spinoza is just as much in earnest in regard to the typical character of mathematics. Spinoza, on the other hand, rigorously carried out the geometrical method, even in externals. To these four principal parts are further added as less essential, deductions or corollaries immediately resulting from the theorems, and the more detailed expositions of the demonstrations or scholia.
Even in speculations on military subjects it was his habit to neglect maps and diagrams, and to put the theorems to practical proof on the ground itself.
But workday experience and its exigencies are stubborn things; and in this process of assimilation of these alien conceptions of right and honest living, it is the borrowed theorems concerning civic rights and duties that have undergone adaptation and revision, not the concrete system of ways and means in which these principles, so accepted, are to be put in practice.
His last remark is, that these dictates of reason are improperly called laws, because 'law, properly, is the word of him that by right hath command over others. But when considered not as mere conclusions or theorems concerning the means of conservation and defence, but as delivered in the word of God, that by right commands all, then they are properly called laws.
Moreover, people's preferences, as evidenced by their choices and decisions in carefully controlled experiments, are inconsistent. This makes most people both irrational and unpredictable. Either one cannot design an experiment to rigorously and validly test theorems and conjectures in economics or something is very flawed with the intellectual pillars and models of this field.
Lastly, Archimedes tells us that Democritus was the first to state, though he could not give a rigorous proof, that the volume of a cone or a pyramid is one-third of that of the cylinder or prism respectively on the same base and having equal height, theorems first proved by Eudoxus.
There can be no discourteous presumption, therefore, in accepting the general theorems of current political theory without prejudice, and looking past the received theoretical formulations for a view of the substantial grounds on which the governmental establishments have grown into shape, and the circumstances, material and spiritual, that surround their continued working and effect.
There are laws or formulæ for guiding the comparison; but the only ones which do not come under the principles of Induction already discussed, are the mathematical axioms of Equality, Inequality, and Proportionality, and the theorems based on them.
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