"The less forcible and lively are commonly denominated Thoughts or Ideas. The other species want a name in our language, and in most others; I suppose, because it was not requisite for any, but philosophical purposes, to rank them under a general term or appellation. Let us, therefore, use a little freedom, and call them Impressions; employing that word in a sense somewhat different from the usual. By the term impression, then, I mean all our more lively perceptions, when we hear, or see, or feel, or love, or hate, or desire, or will. And impressions are distinguished from ideas, which are the less lively perceptions, of which we are conscious, when we reflect on any of those sensations or movements above mentioned." http://www.philosophy-index.com/hume/enquiry-human-understanding/ii.php

Question: I don't understand how there can be intuitive ideas such as geometry when he says all ideas are reflections of impressions we have sensed?


It's not explicitly implied in the exert that all ideas are reflections of collective impressions. Impressions are built upon sense perceptions and feelings. So there are impressions (matters of fact) and there are ideas (relations of ideas), of which are distinctly different. Impressions are unreliable, ideas like Maths aren't.

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All ideas, according to Hume, originate in sense perception. All the basic ideas, according to him, are merely weak copies of sense impressions.

What makes sciences like logic and arithmetic special, by Hume, is that they deal with relations among ideas. They do not however deal with special kinds of ideas.

Hume saw a fundamental difference between arithmetic and geometry. Arithmetic is, according to him, perfectly exact, dealing as it is with relations of ideas. Geometry, on the other hand, is based merely on empirical inferences from sense impressions. Still, Geometry is based, according to Hume, on sense impressions in a way that makes it especially reliable.

Following is an excerpt, where Hume discusses the origin of geometrical judgments.

I have already observed , that geometry, or the art, by which we fix the proportions of figures; though it much excels both in universality and exactness, the loose judgments of the senses and imagination; yet never attains a perfect precision and exactness. It's first principles are still drawn from the general appearance of the objects; and that appearance can never afford us any security, when we examine, the prodigious minuteness of which nature is susceptible. Our ideas seem to give a perfect assurance, that no two right lines can have a common segment; but if we consider these ideas, we shall find, that they always suppose a sensible inclination of the two lines, and that where the angle they form is extremely small, we have no standard of a right line so precise as to assure us of the truth of this proposition. It is the same case with most of the primary decisions of the mathematics. (http://www.philosophy-index.com/hume/treatise-human-nature/1-iii-i.php)

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