"A definite Integral always presupposes numeric values; consequently equations in which definite integrals occur are seldom or never correct as general (formal) equations, but can only be admitted as numeric equations; consequently the convergence of any infinite series which may occur in them is an indispensable condition, whereas the condition of convergence with respect to a general series in general investigations, such as must be necessarily first established as the foundation of the possibility of any calculation, is quite as absurd [...]" (Martin Ohm, "The Spirit of Mathematical Analysis and its Relation to a Logical System", 1842)
"As long, therefore, as an infinite series is still conceived as general, we cannot speak about its divergence or convtergence, precisely because, according to what has gone before, these latter ideas can only make their appearance in conjunction with numerical series (that is, when the general series are expressly considered as numerical)." (Martin Ohm, "The Spirit of Mathematical Analysis and its Relation to a Logical System", 1842)
"Divergent series are in general very mischievous affairs, and it is shameful that any one should have founded a demonstration upon them. You can demonstrate anything you please by employing them, and it is they who have caused so much misfortune, and given birth to so many paradoxes." (Martin Ohm, "The Spirit of Mathematical Analysis and its Relation to a Logical System", 1842)
"It is a remarkable fact that complaints of the want of clearness and rigour in that part of Mathematics which respects calculation, - whether it be called 'Arithmetic', Universal Arithmetic', 'Mathematical Analysis', or aught else, - recur from time to time, now uttered by subordinate writers, now repeated by the most distinguished of the learned. One finds contradictions of the theory of 'opposed magnitudes'; - another is merely disquieted by 'imaginary quantities'; - a third finally meets difficulties in 'infinite series', either because Euler and other distinguished mathematicians have applied them with success in a divergent form, while the complainant thinks himself convinced that their convergence is a fundamental condition, - or because in general investigations general series occur, which, precisely because they are general, can be neither accounted divergent nor convergent. ... The [...] question: how may the paradoxes of calculation be most securely avoided? - obliges us to submit to a very exact examination of the subject of mathematical analysis, its first and simplest ideas, as also the methods of reasoning which are applied to it." (Martin Ohm, "The Spirit of Mathematical Analysis and its Relation to a Logical System", 1842)
"Thus the object of mathematical analysis is perhaps in all cases nothing more than the comparison of magnitudes, but it is totally repugnant to the views of the author to say: 'Mathematics' (and therefore mathematical analysis as a portion of the same) ''is the doctrine of magnitudes (quantities)'. On the contrary, the author has found himself forced to conceive the nature of aaathemadcal analysis much more abstractly^ and he believes Ihat he is much nearer the truth in asserting that: 'mathematical analysis is the doctrine of the relation of ttiose (seven) (mental) acts to one another, to which we are led by the consideration of (whole, indenominate) number', i.e. therefore 'the doctrine of the oppositions' and relations (combinations) in which the above named mental operations stand to (with) one another." (Martin Ohm, "The Spirit of Mathematical Analysis and its Relation to a Logical System", 1842)
"In the most diverse phenomena of the calculus (of arithmetic, algebra, analysis, etc.) the author sees, not properties of quantities, but properties of the operations, that is to saya actions of the understanding [...] It turns out that one only calculates with 'forms', that is, with symbolized operations, actions of the understanding that have been suggested ... by the consideration of the abstract whole numbers." (Martin Ohm, 1855)
" [...] ... he can very well recollect the time when on its first appearance he hardly escaped being declared 'insane' by several mathematicians on account of his views; when he was even designated in official papers as a 'dangerous' innovator on account of these 'revolutionary' ideas of science, and most persons contented themselves either with a silent shrug of the shoulders, or with publicly accusing him of 'presumption'. This public resistance only drove the author to test and retest his views continually, and if possible more rigorously, whereby his works have received a better finish." (Martin Ohm)
"The great clearness and precision manifested in these writings, and their extreme simplicity and logical accuracy, made a forcible impression on the mind of the Translator while pursuing his own mathematical studies as few years ago, and he could not help contrasting these Treatises with the vague, half-elaborated works in his own language. Few persons have indeed pursued the study of Mathematical Analysis with the same anxiety and power to improve the foundations upon which it rests, as Professor Ohm. A life continually spent in instructing other has enabled him to test and retest his views by the best touchstones - the mind of the learner." (Alexander J Ellis [translator of "The Spirit of Mathematical Analysis and its Relation to a Logical System"] 1843)
Resources:
Martin Ohm (1842) "The Spirit of Mathematical Analysis and its Relation to a Logical System" [link]
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