Leonardo Da Vinci spoke on the infinite and the finite, and nothing, as well as that which exists in the mind, or not. His writings pertain to a wide subject matter while exhibiting not only mastery, but creative originality of a high calibre.
In one quote, he states, “A point is not part of a line.” As in, one must break apart the meaning of the infinite, the finite, the coterminous, the mind, the natural, the mathematical, the empirical, and more. This statement reflects the fundamental philosophy of Da Vinci, of which its remnants, as which remained as principles of thought, weave through the quoted works together as one, as in a unified framework for looking at the world, centuries ahead of his time, and now, and hidden to naked eyes, not to the mind.
He said, “It is the infinite alone that cannot be attained, for if it could it would become finite.” In this, the finitude of an existence comes in stark contrast to the infinite. The infinite of which no finite, no matter the size or the number of combinations, could ever match in magnitude.
In his natural philosophy of thought, he comprised a series of independent, unique considerations of the nature of nature, and, by derivative formulation, the nature of human nature vis-à-vis human thought. The finite in contemplation of the infinite.
Da Vinci stated, “The smallest natural point is larger than all mathematical points, and this is proved because the natural point has continuity, and any thing that is continuous is infinitely divisible; but the mathematical point is indivisible because it has no size.”
His point of a smallest “natural” point is in contrast to a smallest point because a “natural” smallest point would include the natural – the real – world of sense and experience as given by the five senses with the most important, to Da Vinci, of sight.
In some manner, in the first quote, he posits the impossibility of an infinite as something attainable, which retains a quality of impossibility for the finite to attain it, instead of the possibility of the existence of the infinite.
In this sense, he merely posits a mathematical truth in the form of the infinite never reachable through the finite. Furthermore, in the personal notebooks, or work books, Da Vinci postulated more.
His premises on a natural point is continuity, so infinity through limitless divisibility, and the “mathematical point” or non-natural point becomes indivisible “because it has no size.” This natural versus mathematical split came firmly to the grasp of the mind of Da Vinci.
He didn’t write carelessly. He was focused and sure of the word as he was in his stroke of the paint brush. The focused separation become the limit versus the limitless, and the natural versus the mathematical.
He comes as a natural philosopher or a scientist, and a mathematician, and so, in both, an empiricist-logician examining for functional relations between things of the mind and things of the sense, so, truly, a logical-operationalist or someone in search of the self-consistent, inside and out, and for functionality, operational truths about the world and the mind.
A logical operationalist, or a self-consistency-operationalist rather more precisely, as one who finds the consistencies of the mind and the world in which one inhabits the evident self-consistent operations in the natural world given by experience and the self-consistent operations of mind in the mathematical world.
He said, “Nothing is that which fills no space. If one single point placed in a circle may be the starting point of an infinite number of lines, and the termination of an infinite number of lines, there must be an infinite number of points separable from this point, and these when reunited become one again; whence it follows that the part may be equal to the whole.”
Nullity, total absence of space; to be a-spatial is to not be, to Da Vinci, where the only existence given by “to be” simply inheres in the language and represents a limit of the language, not of the intrinsic quality or valuation of the original thought of spacelessness as a fundamental premise for nothingness. That is, non-spatial upper limit, lower limit, range, contents, and existence, equate to proper no-things, nothing.
Da Vinci noted the single point in a circle may be the basis for infinity of lines and its coterminous limit with the non-infinite & non-finite, while an infinite separativity exists for the lines themselves with origin in this central point of the circle. His conclusion: the part may be equal to the whole, not is (necessarily) equal to the whole.
In this sense, he derives a principle of reflective capacities of parts to the whole via spatial relations on the premise of an infinite divisibility in actuality and the infinite divisibility permissive of an infinitude of connections from the point in the circle to the rest of the points, as such, in the circle.
On space, once more, Da Vinci, states, “The point, being indivisible, occupies no space. That which occupies no space is nothing. The limiting surface of one thing is the beginning of another.”
Thus a direction of attention to the idea of a point in space, something in a space, mentally, while occupying no space, by definition, causa mentale, becomes something of the mind, not of the world, so equal to no space as occupying no space, so being nothing, as per the derivations before.
While concluding, the limits of the surface of one becomes the beginning of a surface of the other. This raises further questions about the separation of one surface to another, one object to another, where he posits a sort of bleeding of surface to surface as being the nature of the surface of objects, hence the nature of objects derived from direct experience with a property in the object from a subject in relation to the perception of the object in Nature.
These separations of one surface into another formulate something akin to distinctions of mind and not distinctions of nature, so a distinction with a requisite operator on the other side, so Da Vinci himself.
“That which has no limitations, has no form. The limitations of two conterminous bodies are interchangeably the surface of each. All the surfaces of a body are not parts of that body,” Da Vinci continued.
Infinite in all relevant capacities creates no form; thus, form requires finites, where finites exist in the world of the natural in different ways, where limitations exist in “two coterminous bodies” with interchangeability of bodies through surfaces.
In some sense, the properties apart from the person do not reflect the boundaries as in the mind of the experiencer and theorizer, where the coterminous become the partially co-spatial in experience and in mind.
The premise of the limitless meaning no form translates into the self-limiting as that which has form. Any form becomes an intrinsic self-limit on the infinite and, therefore, a finite; finite means form, and infinite means formless.
The surfaces of the object or the “body” become co-spatially extant with the surfaces of the other object(s) or ‘bodies’ in which one becomes the other while separativity remains crucial to an experiential distinction of the objects in perception, in and of themselves.
Da Vinci stated, “The line has in itself neither matter nor substance and may rather be called an imaginary idea than a real object; and this being its nature it occupies no space. Therefore an infinite number of lines may be conceived of as intersecting each other at a point, which has no dimensions and is only of the thickness (if thickness it may be called) of one single line.”
To have neither matter nor substance, or to be “an imaginary idea” rather “than a real object” creates a unitary distinctiveness in the thinking of Da Vinci, where the mind limits the actual in conception, while perception provides strong approximations within the remit of the experience, so providing knowledge of the world through (flawed) direct perception, the objects derived from real objects or the ideas of the real objects amount to the imaginary, where by “its nature occupies no space” so equates to nothing.
An infinity of lines between points in the object conceived in potentiality. Each intersecting in all parts conceived or potentially conceived with a dimensionlessness inherent in the lines, the dots, the intersections, so the object in mind or imaginary idea, itself, as something without dimensions – so being nothing as a-spatial, because dimensionless (non-dimensionality) in mind.
He continued, “The boundaries of bodies are the least of all things. The proposition is proved to be true, because the boundary of a thing is a surface, which is not part of the body contained within that surface; nor is it part of the air surrounding that body, but is the medium interposted between the air and the body, as is proved in its place.”
The most minimal natural boundaries come from the boundaries of “bodies” or objects, such is the phenomenological fact of perception. Where, the boundary is the surface; while the surface is not part of the body “contained within that surface,” and not part of the medium between surfaces, it’s “in place.”
The objects as two spatial volumes; the surfaces as two others; the medium as a fifth separating the two surfaces, then the two surfaces further separating the objects within the direct sensory perception and conception of Da Vinci.
To him, direct experience of the world was the most important. In these quotations, he exhibits a form of experienced ratiocination taking the sensory information from the world, reasoning about it, then deriving principles about both the world and the capacity for personal observation about the world.
A separation between objects, surfaces, and mediums, and the degree of finitude and divisibility within the world of experience and then indivisibility and nothingness within the world of mind, while directing attention to the self-consistency principles in either and the operations functioning behind both and to derive both in direct experience of the world and in the mind, respectively.
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