Remarks on mathematical or demonstrative reasoning:its connexion with logic [&c.].J. Green, 1837 - 135 sider |
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Side 6
... things ; nor to inquire whether the definitions of any particular treatise or mathematician , from Euclid and his nume- rous editors downwards to Newton and his successors , are in every respect the best possible , such as suit best the ...
... things ; nor to inquire whether the definitions of any particular treatise or mathematician , from Euclid and his nume- rous editors downwards to Newton and his successors , are in every respect the best possible , such as suit best the ...
Side 9
... things which have no real ex- istence , and consequently no practical value . I by no means imply that in mathematics the student is to begin with submitting to authority , and not to think about the meaning of the language he uses ...
... things which have no real ex- istence , and consequently no practical value . I by no means imply that in mathematics the student is to begin with submitting to authority , and not to think about the meaning of the language he uses ...
Side 11
... thing signified , ( more simply , the signification , ) which alone is to be pre- sent to the mind in its subsequent appli- cation of the term . The definitions of geometry concern , it is obvious , the meaning of the terms point , line ...
... thing signified , ( more simply , the signification , ) which alone is to be pre- sent to the mind in its subsequent appli- cation of the term . The definitions of geometry concern , it is obvious , the meaning of the terms point , line ...
Side 14
... thing intended , —the object of thought . Much of common reasoning is reasoning from hypothesis in this sense ; that is , it consists in supposing certain re- lations to exist , and in showing that certain consequences follow . It was ...
... thing intended , —the object of thought . Much of common reasoning is reasoning from hypothesis in this sense ; that is , it consists in supposing certain re- lations to exist , and in showing that certain consequences follow . It was ...
Side 15
... things in reality as points , lines , triangles , circles , and squares in the mathematical sense . But if it be so , even admitting all this , still let us remember that the hypotheses or assump- tions , so far as the definitions are ...
... things in reality as points , lines , triangles , circles , and squares in the mathematical sense . But if it be so , even admitting all this , still let us remember that the hypotheses or assump- tions , so far as the definitions are ...
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Vanlige uttrykk og setninger
abstraction admitted agreement or disagreement algebra analysis applied argument Aristotle assent attention called character circle clear cogency common reasoning conceptions conclusion connexion definitions demonstrative reasoning Dissertation distinct Dugald Stewart Edinburgh reviewer Encyclopædia Britannica equal Essay ethics Euclid evidence exact faculties feeling figure and quantity geometrical reasoning geometry Hartley human hypothesis ideas of figure importance inquiry intuitive knowledge knowledge language Laplace lative laws of thought Locke logic magnitudes mathe mathematical reasoning mathematical science mathematical studies matical reasoning matter meaning measure or test ment mental metaphysical middle term mind modes moral moral constitution Natural Philosophy nature nexion notions number and figure object observations peculiar perceive philosophy physical science Playfair premises principles proof proposition reader remarks rience says sense sensible impressions simple ideas Sir James soning straight line student syllogism tain term Logic things signified tical tion treatise triangle true truth Whately Whately's Whewell words writers
Populære avsnitt
Side iii - Read not to contradict and confute, nor to believe and take for granted: But to weigh and consider.
Side 47 - In this case then, when the mind cannot so bring its ideas together, as by their immediate comparison, and as it were juxta-position or application one to another, to perceive their agreement or disagreement, it is fain, by the intervention of other ideas (one or more, as it happens) to discover the agreement or disagreement which it searches ; and this is that which we call reasoning.
Side 18 - In this place we are concerned with nominal definitions only, (except, indeed, of logical terms,) because all that is requisite for the purposes of reasoning (which is the proper province of Logic) is, that a term shall not be used in different senses : a real definition of any thing belongs to the science or system which is employed about that thing.
Side 54 - From this general contrast it will easily be seen, how an excessive study of the mathematical sciences not only does not prepare, but absolutely incapacitates the mind, for those intellectual energies which philosophy and life require.
Side 51 - ... practice, or that even if it had not, it might not still be a dignified and interesting pursuit. One of the chief impediments to the attainment of a just view of the nature and object of logic, is the not fully understanding, or not sufficiently keeping in mind, the SAMENESS of the reasoning process in all cases.
Side 12 - A plane rectilineal angle is the inclination of two straight lines to one another, which meet together, but are not in the same straight line.
Side 46 - The next degree of knowledge is, where the mind perceives the agreement or disagreement of any ideas, but not immediately.
Side 32 - In any right-angled triangle, the square which is described on the side subtending the right angle is equal to the squares described on the sides which contain the right angle.
Side 47 - Now, in every step reason makes in demonstrative knowledge, there is an intuitive knowledge of that agreement or disagreement it seeks with the next intermediate idea, which it uses as a proof : for if it were not so, that yet would need a proof; since without the perception of such agreement or disagreement there is no knowledge produced. If it be perceived by itself, it is intuitive knowledge : if it cannot be perceived by itself, there is need of some intervening idea, as a common measure, to...
Side 47 - Those intervening ideas which serve to show the agreement of any two others, are called proofs; and where the agreement or disagreement is by this means plainly and clearly perceived, it is called demonstration, it being shown to the understanding, and the mind made to see that it is so.