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as evident as any thing whatsoever can be*. You may bring the matter to the teft of the fenfes, by laying a few halfpence and farthings upon the table; but
Different opinions have prevailed concerning the nature of thefe geometrical axioms. Some fuppofe, that an axiom is not felf-evident, except it imply an identical propofition; that therefore this axiom, It is impofible for the fame thing, at the fame time, to be and not to be, is the only axiom that can properly be called intuitive; and that all thofe other propofitions commonly called axioms, ought to be demonftrated by being refolved into this fun'damental axiom. But if this could be done, which I fear is not poffible, mathematical truth would not be one whit more certain than it is. Thofe other axioms produce abfoluté certainty, and produce it immediately, without any procefs of thought or reafoning that we can difcover. And if the truth of a propofition be clearly and certainly perceived by all men without proof, and if not proof whatfoever could make it more clear or more certain, it feems captious not to allow that propofition the name of Intuitive Axiom. Others fuppofe, that though the demonftration of mathematical axioms is not abfolute. ly neceffary, yet that thefe axioms are fufceptible of demonftration, and ought to be demonftrated to those who require it. Dr Barrow is of this opinion. So is Apol
attempted a demon
equal to one and the
lonius; who, agreeably to it, has ftration of this axiom, That things fame thing are equal to one another.- But whatever ac count we make of thefe opinions, they affect not our doctrine. However far the demonftration of axioms, may be carried, it mult at laft terminate in one principle of common fenfe, if not in many; which principle we must take for granted whether we will or not.
the evidence of fenfe is not more unqueftionable, than that of abstract intuitive truth; and therefore the former evidence, though to one ignorant of the meaning of the terms, it might serve to explain and illuftrate the latter, can never prove it. But not to rest any thing on the fignification we affix to the word proof, and to remove every poffibility of doubt as to this matter, let us fuppofe, that the evidence of external fenfe is more unquestionable than that of abftract intuitive truth, and that every intuitive principle in mathematics may thus be brought to the test of fenfe; and if we cannot call the evidence of fenfe a proof, let us call it a confirmation of the abstract principle: yet what do we gain by this method of illustration? We only difcover, that the evidence of abftract intuitive truth is refolvible into, or may be illuftrated by, the evidence of fenfe. And it will be feen in the next fection, that we believe in the evidence of external fenfe, not because we can prove it to be true, but because the law of our nature determines us to believe in it without proof. So that in whatever way we view this fubject, the point we propose to illuftrate
illuftrate appears unquestionably certain, namely, "That all mathematical truth is "founded in certain firft principles, which (પ common fenfe or inftinct compels us to "believe without proof, whether we will <6 or not."
Nor would the foundation of mathematics be in the leaft degree more ftable, if thefe axioms did admit of proof, or were all refolvible into one primary axiom expreffed by an identical propofition. Aş the cafe now ftands, we are abfolutely certain of their truth; and abfolute certainty is the utmost that demonftration can produce. We are convinced by a proof, because our conftitution is fuch, that we must be convinced by it: and we believe a felf-evident axiom, becaufe our conftitution is fuch that we must believe it.. You afk, why I believe what is felf-evident? I may as well afk, why you believe what is proved? Neither question admits of an anfwer; or rather, to both questions the anfwer is the fame, namely, Because I muft believe it.
Whether our belief in thefe cafes be agreeable to the eternal relations and fitneffes of things, and fuch as we should entertain
entertain if we were perfectly acquainted with all the laws of nature, is a queftion which no perfon of a found mind can have any fcruple to anfwer, with the fulleft af furance, in the affirmative. Certain it is, our conftitution is fo framed, that we must believe to be true, and conformable to univerfal nature, that which is intimated to us by the original fuggeftions of our own understanding. If thefe are fallacious, it is the Deity who makes them fo; and therefore we can never rectify, or even. detect, the fallacy. But we cannot even fuppofe them fallacious, without violating our nature; nor, if we acknowledge a God, without the moft abfurd and most audacious impiety; for in this fuppofition it is implied, that we fuppofe the Deity a deceiver. Nor can we, confiftently with fuch a fuppofition, acknowledge any diftinction between truth and falfehood, or believe that one inch is lefs than ten thousand miles, or even that we ourfelves exift.
Of the Evidence of External Senfe.
A Nother clafs of truths producing con
viction, and abfolute certainty, are thofe which depend upon the evidence of the external fenfes; Hearing, Seeing, Touching, Tafting, and Smelling. On this evidence depends all our knowledge of external or material things; and therefore all conclufions in Natural Philofophy, and all those prudential maxims which regard the prefervation of our body, as it is liable to be affected by the fenfible qualities of matter, must finally be refolved into this principle, That things are as our fenfes represent them. When I touch a ftone, I am confcious of a certain fenfation, which I call a fenfation of hardness. But this fenfation is not hardness itself, nor any thing like hardness it is nothing more than à fenfation or feeling in my mind; accompanied, however, with an irresistible belief, that this fenfation is excited by the application of an external and hard fubftance