"Matters of fact, which are the second objects of human reason, are not ascertained in the same manner, nor is an evidence of their truth, however great, of a like nature with the foregoing. The contrary of every matter of fact is still possible, because it can never imply a contradiction, and is conceived by the mind with the same facility and distinctness, as if ever so conformable to reality. _That the sun will not rise to-morrow_, is no less intelligible a proposition, and implies no more contradiction, than the affirmation, _that it will rise_. We should in vain, therefore, attempt to demonstrate its falsehood. Were it demonstratively false, it would imply a contradiction, and could never be distinctly conceived by the mind."--(IV. pp. 32, 33.) The distinction here drawn between the truths of geometry and other kinds of truth is far less sharply indicated in the _Treatise_, but as Hume expressly disowns any opinions on these matters but such as are expressed in the _Inquiry_, we may confine ourselves to the latter; and it is needful to look narrowly into the propositions here laid down, as much stress has been laid upon Hume's admission that the truths of mathematics are intuitively and demonstratively certain; in other words, that they are necessary and, in that respect, differ from all other kinds of belief. What is meant by the assertion that "propositions of this kind are discoverable by the mere operation of thought without dependence on what is anywhere existent in the universe"? Suppose that there were no such things as impressions of sight and touch anywhere in the universe, what idea could we have even of a straight line, much less of a triangle and of the relations between its sides? The fundamental proposition of all Hume's philosophy is that ideas are copied from impressions; and, therefore, if there were no impressions of straight lines and triangles there could be no ideas of straight lines and triangles. But what we mean by the universe is the sum of our actual and possible impressions. So, again, whether our conception of number is derived from relations of impressions in space or in time, the impressions must exist in nature, that is, in experience, before their relations can be perceived. Form and number are mere names for certain relations between matters of fact; unless a man had seen or felt the difference between a straight line and a crooked one, straight and crooked would have no more meaning to him, than red and blue to the blind. The axiom, that things which are equal to the same are equal to one another, is only a particular case of the predication of similarity; if there were no impressions, it is obvious that there could be no predicates. But what is an existence in the universe but an impression?