Hi All,

    I'd like to offer some thoughts on Spinoza's usage of examples and illustrations from mathematics, including geometry, for Reason and Intuition in his works as these seem to present a problem for some people. I don't believe that his choice of the various mathematical examples was made in haste, and they are used in virtually all of his writings, so I don't think he had any second thoughts about them and he certainly was well aware that they were dealing with abstractions (see example below.) Regardless of this, I have no doubt that many people wish he had not used any such things at all, just as many think it was an utter mistake on his part to write the Ethics in the "geometrical fashion." He even writes in Letter 2; Spinoza to Oldenburg, concerning a particular point of discussion; "...I can think of nothing better than to submit them to the bar of your judgment proved in the geometrical method."

    Spinoza uses the same example we are now looking at (given three numbers, to find the fourth number proportional, which will be to the third as the second is to the first), for illustrating the different Kinds of Knowledge, in three different works; "The Improvement", "The Short Treatise", and "The Ethics". He refers to Euclid's Elements, Book 7, Prop. 19 to illustrate the kind of knowledge he names Reason, and which kind of knowledge he uses and expects his readers to follow and affirm with their own mind nearly everywhere in his writings. I suppose that by the time he decided, and actually went to publish the Ethics in 1675 (according to Letter 19(68) - Spinoza to Oldenburg; where he also explains why he backed off from going through with the plan), just a little more than a year and a half before he died, he had had plenty of time to come up with something better if he felt there was a need for it. However, apparently he felt that he had illustrated his point in a very clear way.

    He also uses the example of the nature of a triangle being such that its three interior angles are equal to two right angles (which he says in Letter 60 (56), Spinoza to Hugo Boxel, that he learned from Euclid's Elements [see Eulcid's Elements, Book 1, Prop. 32]), in many different writings, including "The Improvement", "The Short Treatise", "The Theological-Political Treatise", and "The Ethics", and also in several personal letters. In particular, in the treatise being discussed here, he writes:

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...until we reflect on the knowledge which we have of the first principle of all things, and find that which teaches us that God is not a deceiver, and until we know this with the same certainty as we know from reflecting on the nature of a triangle that its three angles are equal to two right angles. But if we have a knowledge of God equal to that which we have of a triangle, all doubt is removed....
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    Is he not saying that we might have the "same certainty" concerning "knowledge of God" as we might also have of the triangle?! I imagine of course that there are some people who haven't the slightest idea as to why, or even if; "the three interior angles of a triangle are equal to two right angles", and they would have preferred if he would have just said something like; "as sure as I'm standing here", or maybe; "as sure as I know that I 'freely' chose to write this post" :-), but he used an idea which he apparently felt that anyone willing to put in the effort would find their own mind affirming as an eternal truth.

    Spinoza uses another example from Euclid in the Ethics to illustrate a point, as far as possible, in the Note following E2P8C. Although he does not explicitly refer to it there, the proof of the "equal rectangles" contained in a circle that he uses is found in Euclid, Book 3, Prop. 35.

    And, as we will find out a bit later in "The Improvement", he uses other simple geometric ideas, such as the following, to make his point. Note that he says he is purposely choosing something "abstract" so I hope no one thinks that they can just ignore and skip over anything in Spinoza's writings which they don't understand and just say to themselves; "Well, this isn't an 'Intuitive Idea', it's 'abstract' and deals with 'beings of reason' so it has no real value anyway.":

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    A definition, if it is to be called perfect, must explain the inmost essence of a thing, and must take care not to substitute for this any of its properties. In order to illustrate my meaning, without taking an example which would seem to show a desire to expose other people's errors, I will choose the case of something abstract, the definition of which is of little moment. Such is a circle. If a circle be defined as a figure, such that all straight lines drawn from the center to the circumference are equal, every one can see that such a definition does not in the least explain the essence of a circle, but solely one of its properties. Though, as I have said, this is of no importance in the case of figures and other abstractions, it is of great importance in the case of physical beings and realities: for the properties of things are not understood so long as their essences are unknown....

...If the thing in question be created, the definition must (as we have said) comprehend the proximate cause. For instance, a circle should, according to this rule, be defined as follows: the figure described by any line whereof one end is fixed and the other free. This definition clearly comprehends the proximate cause.
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...and, as for any belief that all these geometric examples are referring to the particular image which might form in your brain as you read the above or which you might look at in a book or draw for yourself, those are "images", not the "ideas" Spinoza is referring to. See for example:

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...For instance, in order to form the conception of a sphere, I invent a cause at my pleasure--namely, a semicircle revolving round its centre, and thus producing a sphere. This is indisputably a true idea; and, although we know that no sphere in nature has ever actually been so formed, the perception remains true, and is the easiest manner of conceiving a sphere....
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    He says, "...This is indisputably a true idea...", even though; "...no sphere in nature has ever actually been so formed,...", so even if we are looking at a diagram or an animation which seems to show such a sphere being created in that manner, he is telling us that it is the "true idea" that he is referring to, not some object of our senses or some imagined thing which is only a modification of our own body (and it's not the idea of that particular bodily modification either. In the Ethics Spinoza will show that the Understanding is Eternal and does not depend on the existence of the body although those particular images do.)

    I believe that this will be hard for some to grasp but Spinoza will go over this again and again, that we must distinguish between "words and images", which are put together from the motion and rest of our own body, and "ideas", which are affirmations of the mind and which themselves, as ideas, involve no motion or rest. See for instance:

====== Ethics E2P49C Note:
...an idea, being a mode of thinking, does not consist in the image of anything, nor in words.The essence of words and images is put together by bodily motions, which in no wise involve the conception of thought....
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    As for Intuition, Spinoza offered two examples, both from simple mathematics, that I believe anyone MIGHT understand but which may also be confused with the images involved (and therefore NOT be understood), so try to realize that although there will undoubtedly be some image formed in your brain as you read and think about this, those images are not what Spinoza is referring to.:

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...By the same kind of knowledge we know that two and three make five, or that two lines each parallel to a third, are parallel to one another, etc....
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    "Two and three make five." He is not referring to the image of two apples and of three apples taken together which our imagination may offer up, nor to the numbers we might write out such as: 2 + 3 = 5. He is talking about the affirmation of "twoness" and "threeness" and "fiveness", etc., again, not about any words or images that may also form in our brain.

    The same goes for; "two lines each parallel to a third, are parallel to one another." And again, he is using these as examples of what he means by Intuition (direct knowledge, no process.) In his other illustration, regarding proportion, he said that he knows (at least for simple numbers as he writes in the Ethics); "without going through any process."

    Are these things abstract? If it matters then why did he use them as illustrations of Intuition? He is only dealing at this point with the nature of the kinds of knowledge in preparation for the more important things to come but he is trying to lead us there by these steps (at least if we are going to use his method.)

    As an exercise in clear thinking (does not Spinoza actually suggest this by using the examples from Euclid? or should we suppose that he was just trying to appeal to some external authority to lend credence to his own ideas?), you might take a look at Euclid's Elements, although, as with the Ethics, you may have trouble if you just dive into it anywhere since it builds progressively from definitions, axioms, and previously proven propositions. I believe, for instance, that if anyone who has not done so would put in the effort to understand Euclid's proof (mentioned above), concerning the sum of the interior angles of a triangle (it is near the beginning of the work but does refer back to a few previous propositions which you may need to study over too), they would then have an excellent example in their own mind and a real "taste" for what Spinoza means by certainty of this kind, namely Reason. In this way your own mind may begin to affirm the truth of what you have been accepting as hearsay --even if it was hearsay from Spinoza himself. Is it not our own mind which we are seeking to Improve?

    Spinoza uses Reason both here in "The Improvement", and of course in the Ethics and elsewhere, so if we were hoping to find some short cut to Blessedness we might need to look elsewhere than in Spinoza's writings:

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    It may, perhaps, provoke astonishment that, after having said that the good method is that which teaches us to direct our mind according to the standard of the given true idea, we should prove our point by reasoning, which would seem to indicate that it is not self-evident....

...for proving the truth, and for valid reasoning, we need no other means than the truth and valid reasoning themselves: for by valid reasoning I have established valid reasoning, and, in like measure, I seek still to establish it....
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    Best Regards,
        Terry