If it is not intuited, then how do you know that one thing follows from another? — John
As I said, I really don't know what you mean by intuitive. My dictionary defines intuition as "immediate apprehension by the mind without reasoning". There is a second definition which is immediate apprehension by a sense, and a third definition which is immediate insight. Notice that all use the word "immediate". Logic is a tool which the mind uses, so there is a necessity to understand the premises, the principles of logic being employed, and how these are related, so that a conclusion is drawn. There is no immediacy here, the conclusion requires mental effort, so it is impossible that logical conclusions are "intuitive", instead, they are "rational".
The form of maple leaves for example is a general form that can only be grasped as a visualization, it will be a description that will result in a visualization, an algebraic expression which specifies the coordinates on the x/y axis which will result in a visualization, or a direct pictorial representation. — John
Again, I disagree with you. I think that it is impossible to grasp a general form through a visualization. That's the very essence of a "general" form. A visual image is always of a particular, and a general form is categorically different from this. There is an inherent incompatibility between the two. That is why the general is understood through definition, rather than through visualizing a particular. We can come to grasp the general through seeing particular instances, and abstracting certain properties, but we cannot immediately grasp the general from a particular instance. This would merely be grasping a particular, and clearly not a case of grasping the "general form", which requires understanding some general principles.
Now, let's take the example of the pictorial representation: it is probably never going to be the exact form of any particular maple leaf, because each particular maple leaf has its own form of the same general kind which can be abstracted to form a closer, but never perfect, visual representation. — John
Here is the difficulty with your presentation. Unless the representation is of a particular maple leaf, then it is not a visual, or "pictorial" representation at all. A pictorial representation, or imaginary image, is of a particular, even if that particular is only within your mind. The image consists of the particulars which your mind puts there. You can imagine the maple leaf as just a stem, with a vague shape on top, many particulars not filled in by your mind's imagining process, but this does not give you the general form. What it gives you is an incomplete particular.
So what you don't seem to be understanding is that when we abstract the form, from a particular, through sense and thought, it is always a particular form which is abstracted from that individual. The particular form abstracted will be incomplete, depending on which aspects of the form are apprehended as important. But this does not produce a general form. The general form is created by the mind adopting certain principles of recognition, and classification. This is what Aristotle described as distinguishing essentials from accidentals. By employing these principles, first to recognize the similarity between different leaves, and second, to hold that there is a classification called "maple leaf", the existence of a general form is demonstrated. Further, an individual human being is often mistaken in these principle, so the formal existence of a general form is by definition. The classification is named and the defining features are described, such that there is agreement amongst human beings, and the general form maintains its existence by means of this agreement, convention.
The point being made, is that the particular form is what is abstracted from the particular object. This is perception. The general form is something completely different, it is created by the mind. The general form exists as principles of recognition and classification, rules which the mind follows. There is a categorical difference between the two. The two cannot be conflated because there is a deep incompatibility between them, and this is why dualism is necessary.
But there is no sense in which the present form of the maple leaf existed prior to the present moment. Prior to the present moment there were a succession of slightly different forms that evolved to the present form. — John
Let me try once again, to explain this issue. As time passes, there is as you say, "a succession of slightly different forms". At each moment of the present, the maple leaf is this particular maple leaf, it is not that particular maple leaf which it was at the last moment, because it has changed. Therefore at each moment the maple leaf is a new, and different object. So at each moment a new object is created, we can call them MLt1, MLt2, MLt3, etc., each collection of symbols referring to a different object. Let's take MLt3 for example. When that object comes into existence, it necessarily comes into existence as the object which it is, MLt3, or else it is not MLt3. It does not come into existence as MLt2, Mlt4, or any random thing, it comes into existence as MLt3. Therefore we can assume that there is a cause of its existence as MLt3, a reason why it exists at that moment as MLt3, and not something else. This is the determining form of MLt3. Notice that in order for the object, MLt3, to exist at that present moment, as MLt3, it is necessary that the form of MLt3 existed prior to that. This prior form is not MLt2, it is not MLt4, because these are distinctly different. It is nothing other than the form of MLt3, which exists prior to the object MLt3, and ensures that object MLt3 will exist as that object, at that moment in time.
But we see material object MLt3. We abstract that particular form, and this constitutes our representation of object MLt3. Our representation is a representation of material object MLt3, it is not a representation of the determining form, which exists prior to material object Mlt3, ensuring that Mlt3 will exist as MLt3.
The visual form of an object cannot be grasped as a mathematical formula, well at least I can't grasp it as such, and I have spoken to mathematicians who say the same. The form can be modeled as a mathematical formula, the formula can of course be understood in purely mathematical terms, but it cannot be visualized directly, as mathematical formula; by definition it can only be visualized as a visual form. If you still disagree then there is no point continuing, because I am just going to say you are wrong; and you are probably just going to say I am wrong, and it will be a waste of time and energy. — John
The point I am making is that the visual form of the object is distinctly different from the form of the object which precedes the existence of that object in time, causing it to be that object which it is. The visual form of the object is created by human perception following the object's presence in time. It is a representation of the object's material existence at that time. The form which precedes the object's material existence, and determines what that object will be at any moment in time, as time passes, cannot be seen visually, because it is always prior in time to that object's material existence, which is what is seen. The only access we have to this prior form is the mathematical formula, which enables us to predict, and is inherently different from the visual form. The difference, I have argued, is that the visual form is spatial, and the mathematical is non-spatial. This, I believe, is due to the fact that there is no spatial existence prior to the present moment in time. Spatial existence is created at each moment of passing time.
So I am in agreement with you, that the visual form of the object cannot be grasped by the mathematical formula. The two are deeply incompatible, and that's why I advocate dualism. Here's an explanation of this incompatibility:
We see the object, thus creating a visual form. What we see is the object's material form, and this is necessarily post-present in time. The object is present to us, as a material object, at the present, so our representation of it is necessarily post-present, therefore this particular form is post-present. As human beings we proceed to create general forms, these are generalized rules, rules for naming, classifying, right up to the general laws of physics. Then, we turn back to the particular object, applying these general rules. So we have mathematical formulae, which we apply to the particular objects, attempting to determine the pre-present form of the object. This is prediction. But the essential nature of the pre-present form of the object is that it is particular. Each object has its own particular form, proper to it, which is prior to it in time, causing it to be the object which it is. Now we have general forms, mathematical, and physical laws etc., which are not specific to the particulars of the object. So we have an incompatibility between the general nature of mathematical formulae, and the particular nature of the pre-present forms of objects.