Math is empirical

[Anonymous No.22290012]

>If, as laid down in the two preceding chapters, the foundation of all sciences, even deductive or demonstrative sciences, is Induction; if every step in the ratiocinations even of geometry is an act of induction; and if a train of reasoning is but bringing many inductions to bear upon the same subject of inquiry, and drawing a case within one induction by means of another; wherein lies the peculiar certainty always ascribed to the sciences which are entirely, or almost entirely, deductive? Why are they called the Exact Sciences? Why are mathematical certainty, and the evidence of demonstration, common phrases to express the very highest degree of assurance attainable by reason? Why are mathematics by almost all philosophers, and (by some) even those branches of natural philosophy which, through the medium of mathematics, have been converted into deductive sciences, considered to be independent of the evidence of experience and observation, and characterized as systems of Necessary Truth?
>The answer I conceive to be, that this character of necessity, ascribed to the truths of mathematics, and (even with some reservations to be hereafter made) the peculiar certainty attributed to them, is an illusion; in order to sustain which, it is necessary to suppose that those truths relate to, and express the properties of, purely imaginary objects... Now we have pointed out that, from a definition as such, no proposition, unless it be one concerning the meaning of a word, can ever follow; and that what apparently follows from a definition, follows in reality from an implied assumption that there exists a real thing conformable thereto. This assumption, in the case of the definitions of geometry, is not strictly true: there exist no real things exactly conformable to the definitions. There exist no points without magnitude; no lines without breadth, nor perfectly straight; no circles with all their radii exactly equal, nor squares with all their angles perfectly right. It will perhaps be said that the assumption does not extend to the actual, but only to the possible, existence of such things. I answer that, according to any test we have of possibility, they are not even possible... Their existence, so far as we can form any judgment, would seem to be inconsistent with the physical constitution of our planet at least, if not of the universe... The points, lines, circles, and squares which any one has in his mind, are (I apprehend) simply copies of the points, lines, circles, and squares which he has known in his experience. Our idea of a point, I apprehend to be simply our idea of the minimum visibile, the smallest portion of surface which we can see. A line, as defined by geometers, is wholly inconceivable...We can reason about a line as if it had no breadth; because we have... the power, when a perception is present to our senses... of attending to a part only of that perception or conception, instead of the whole.

[Anonymous No.22290017]

>>22290012
>Since, then, neither in nature, nor in the human mind, do there exist any objects exactly corresponding to the definitions of geometry, while yet that science can not be supposed to be conversant about nonentities; nothing remains but to consider geometry as conversant with such lines, angles, and figures, as really exist; and the definitions, as they are called, must be regarded as some of our first and most obvious generalizations concerning those natural objects. The correctness of those generalizations, as generalizations, is without a flaw: the equality of all the radii of a circle is true of all circles, so far as it is true of any one: but it is not exactly true of any circle; it is only nearly true; so nearly that no error of any importance in practice will be incurred by feigning it to be exactly true. When we have occasion to extend these inductions, or their consequences, to cases in which the error would be appreciable—to lines of perceptible breadth or thickness, parallels which deviate sensibly from equidistance, and the like—we correct our conclusions, by combining with them a fresh set of propositions relating to the aberration; just as we also take in propositions relating to the physical or chemical properties of the material, if those properties happen to introduce any modification into the result; which they easily may, even with respect to figure and magnitude, as in the case, for instance, of expansion by heat. So long, however, as there exists no practical necessity for attending to any of the properties of the object except its geometrical properties, or to any of the natural irregularities in those, it is convenient to neglect the consideration of the other properties and of the irregularities, and to reason as if these did not exist: accordingly, we formally announce in the definitions, that we intend to proceed on this plan. But it is an error to suppose, because we resolve to confine our attention to a certain number of the properties of an object, that we therefore conceive, or have an idea of, the object, denuded of its other properties. We are thinking, all the time, of precisely such objects as we have seen and touched, and with all the properties which naturally belong to them; but, for scientific convenience, we feign them to be divested of all properties, except those which are material to our purpose, and in regard to which we design to consider them.

[Anonymous No.22290022]

>>22290017
>The peculiar accuracy, supposed to be characteristic of the first principles of geometry, thus appears to be fictitious. The assertions on which the reasonings of the science are founded, do not, any more than in other sciences, exactly correspond with the fact; but we suppose that they do so, for the sake of tracing the consequences which follow from the supposition. The opinion of Dugald Stewart respecting the foundations of geometry, is, I conceive, substantially correct; that it is built on hypotheses; that it owes to this alone the peculiar certainty supposed to distinguish it; and that in any science whatever, by reasoning from a set of hypotheses, we may obtain a body of conclusions as certain as those of geometry, that is, as strictly in accordance with the hypotheses, and as irresistibly compelling assent, on condition that those hypotheses are true.68

>When, therefore, it is affirmed that the conclusions of geometry are necessary truths, the necessity consists in reality only in this, that they correctly follow from the suppositions from which they are deduced. Those suppositions are so far from being necessary, that they are not even true; they purposely depart, more or less widely, from the truth. The only sense in which necessity can be ascribed to the conclusions of any scientific investigation, is that of legitimately following from some assumption, which, by the conditions of the inquiry, is not to be questioned. In this relation, of course, the derivative truths of every deductive science must stand to the inductions, or assumptions, on which the science is founded, and which, whether true or untrue, certain or doubtful in themselves, are always supposed certain for the purposes of the particular science. And therefore the conclusions of all deductive sciences were said by the ancients to be necessary propositions. We have observed already that to be predicated necessarily was characteristic of the predicable Proprium, and that a proprium was any property of a thing which could be deduced from its essence, that is, from the properties included in its definition.

[Anonymous No.22290028]

>>22290012
if I wanted to read the book you're quoting I'd get it and read it myself separately. Just make your point or stop spamming that shit please. No one wants to read it.

[Anonymous No.22290039]

>>22290028
No one is forcing you to read anything.

[Anonymous No.22290061]

>>22290039
You killed a thread to quote a book and didn't make any point of your own. No one wants to read it.

[Anonymous No.22290090]

>>22290012
Math is the premier example of deductive reasoning. If it is really inductive then there is no deductive reasoning and every field of human knowledge is empirical.

[Anonymous No.22290092]

>>22290061
Relax. Here’s the thread I killed btw >>22288492, if you’re so upset, you can kill mine by remaking it when mine is at the bottom of the catalog.

[Anonymous No.22290098]

>>22290090
I think he’s saying the foundations (axioms and definitions) are inductive. Obviously what you draw from that is deductive.

[Anonymous No.22290155]

>>22290012
Every belief that leads to any useful action is empirical and falsifiable.

[Anonymous No.22290198]

>>22290098
>I think he’s saying the foundations (axioms and definitions) are inductive.
Those can be anything you want. If all he's saying is that the axioms we are interested in are ones that intuitively seem to model reality he's not saying anything new

[Anonymous No.22290275]

>>22290012
>if every step in the ratiocinations even of geometry is an act of induction
This fuckwit hasn't even studied Euclid. Geometry was founded on a series of axioms.
Empirical, my hairy arse.

[Anonymous No.22290583]

>>22290198
No they can’t be “anything you want.” Because there is no way to prove that the set of axioms will be consistent if they don’t have some basis in reality or intuition.

[Anonymous No.22290604]

>>22290275
He read Euclid in the original greek when he was eight years old. You on the other hand haven never read anything that wasn’t posted on 4chan.

[Anonymous No.22290623]

>>22290275
You can’t derive the axioms without empirical experience. The laws of logic are apprehended by the observation of the repetition of such patterns in the world. Perhaps we intuit some understanding, but that is just because are brains evolved to do so.

[Anonymous No.22290758]

>>22290604
Then his idiotic assertion is even less forgivable.
>>22290623
Guessing you have no qualifications whatsoever in either maths or logic. And if you have, you should give them back.

[Anonymous No.22290785]

>>22290758
not an argument