Not only did Peirce defend infinitesimals.
Newton introduced his own theory of infinitely small numbers, or infinitesimals, to justify the calculation of derivatives, or slopes.
of invertible elements of K are naturally identified as invertible infinitesimals.
In mathematics, the best available theory concerning infinitesimals was inconsistent.
The principle of microcancellation supplies the exact sense in which there are “enough” infinitesimals in smooth infinitesimal analysis.
Some of his sharpest criticism was directed at those mathematicians, such as Fermat, who used infinitesimals in the construction of tangents.
But later he came to adopt a more tolerant attitude towards infinitesimals, regarding them as useful fictions in somewhat the same way as did Leibniz.
Since arguments using infinitesimals are usually easier to visualise than arguments using limits, nonstandard analysis is a helpful tool for mathematical analysts.
While Euler treated infinitesimals as formal zeros, that is, as fixed quantities, his contemporary Jean le Rond d'Alembert (1717–83) took a different view of the matter.
Multiplicative inverses of nonstandard integers are infinitesimals, but, being themselves invertible, they are of a different type from the ones we have considered so far.
Leibniz's attitude toward infinitesimals and differentials seems to have been that they furnished the elements from which to fashion a formal grammar, an algebra, of the continuous.
For Leibniz the incomparable smallness of infinitesimals derived from their failure to satisfy Archimedes' principle; and quantities differing only by an infinitesimal were to be considered equal.
In 1748, in his Introductio in analysin infinitorum, he developed the concept of function in mathematical analysis, through which variables are related to each other and in which he advanced the use of infinitesimals and infinite quantities.
Although the use of infinitesimals was instrumental in Leibniz's approach to the calculus, in 1684 he introduced the concept of differential without mentioning infinitely small quantities, almost certainly in order to avoid foundational difficulties.
But behind the formal beauty of these rules—an early manifestation of what was later to flower into differential algebra—the presence of infinitesimals makes itself felt, since Leibniz's definition of tangent employs both infinitely small distances and the conception of a curve as an infinilateral polygon.
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In certain models of SIA the system of natural numbers possesses some subtle and intriguing features which make it possible to introduce another type of infinitesimal—the so-called invertible infinitesimals—resembling those of nonstandard analysis, whose presence engenders yet another infinitesimal neighbourhood of 0 properly containing all those introduced above.
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Peirce's conception of the number continuum is also notable for the presence in it of an abundance of infinitesimals, Peirce championed the retention of the infinitesimal concept in the foundations of the calculus, both because of what he saw as the efficiency of infinitesimal methods, and because he regarded infinitesimals as constituting the “glue” causing points on a continuous line to lose their individual identity.
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Peirce's conception of the number continuum is also notable for the presence in it of an abundance of infinitesimals Peirce championed the retention of the infinitesimal concept in the foundations of the calculus both because of what he saw as the efficiency of infinitesimal methods and because he regarded infinitesimals as constituting the glue causing points on a continuous line to lose their individual identity