ONT Re: Hermeneutic Equivalence Classes
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| Leibniz, "Elements of a Calculus" (cont.)
|
| 5. We shall use letters (such as 'a', 'r', 'h', or 'm', 'p', 's' above) when
| numbers are either not available, or they are at any rate being treated
| generally and not considered specifically. This we must do here, when
| we are establishing the elements of the subject. The same thing is
| done in algebra, so that we are not compelled to show in individual
| cases what we can show once and for all of an indefinite number of
| instances. The method of using letters here I shall explain below.
|
| 6. The rule given in article 4 is sufficient for our calculus to cover
| all things in the whole world, as far as we have distinct concepts
| of them, i.e. as far as we know some of their requisites by which,
| after we have examined them bit by bit, we can distinguish them
| from all others; or, as far as we can assign their definition.
| For these requisites are simply the terms whose concepts compose
| the concept which we have of a thing.
|
| We can distinguish many things from others by their requisites,
| and if there are any whose requisites are difficult to assign,
| we will assign to them in the mean time some prime number,
| and use it to designate other things. [?].
|
| In this way we shall be able to discover and prove
| by our calculus at any rate all the propositions
| which can be proved without the analysis of what
| has temporarily been assumed to be prime. (In
| the same way, Euclid never uses the definition
| of a straight line in his proofs, but instead
| used certain assumptions which he took to be
| axiomatic. But when Archimedes wanted to
| go further, he was compelled to analyse
| and define the straight line itself --
| namely, as the least distance between
| two points.)
|
| In this way we shall discover, if not all, at any rate innumerable things;
| both those which have already been proved by others, and those which can
| ever be proved by others from the definitions, axioms, and experiments
| which are already known.
|
| This is our prerogative: that by means of numbers we can judge immediately
| whether propositions presented to us are proved, and that what others could
| hardly do with the greatest mental labour and good fortune, we can provide
| with the guidance of symbols alone, by a sure and truly analytical method.
| As a result of this, we shall be able to show within a century what many
| thousands of years would hardly have granted to mortals otherwise.
|
| Leibniz, 'Logical Papers', p. 18.
|
| Leibniz, G.W., "Elements of a Calculus" (April, 1679),
| G.H.R. Parkinson (ed.), 'Leibniz: Logical Papers', pp. 17-24,
| Oxford University Press, London, UK, 1966. (Couturat, 49-57).
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