In Part B, we observed that, in the history of human knowing, a major transfiguration occurred within scientific knowing - certain personalities went from patiently observing, thinking about, and documenting the behavior of outer phenomena (sense-perceptions) to inwardly systematizing the phenomenal relations by way of purely ideal mathematical concepts. What do we mean by "transfiguration"? This word is used because it is really the most useful to convey useful meaning to the intellect. We can immediately imagine that it conveys how our thinking can becomes like a new sense-organ, perceiving thought-forms just as our other senses perceive their respective qualities, in a way that we could not before. Is this development not what literally happens? Anyone who has not studied mathematics at all, and then begins to learn it with discpline and effort, can sense how they have attained a mode of perceiving the world content which was not there before. Before we put in that knowing effort, a system of mathematical concepts, like algebra or geometry, will appear to us "like a theory of music to the deaf". Just as Scorates prefigured the future course of 'knowing' in Western civilization, he (and others like Pythagoras) also prefigured its crowning achievement - the development of the mathematical sense-organ. Is this thinking sense-organ the final one for humanity? It is now we really need to recall that all error in modern age is born of incompleteness. Below we will consider an illustration of that:
Steiner wrote:It is quite arbitrary to regard the sum of what we experience of a thing through bare perception as a totality, as the whole thing, while that which reveals itself through thoughtful contemplation is regarded as a mere accretion which has nothing to do with the thing itself. If I am given a rosebud today, the picture that offers itself to my perception is complete only for the moment. If I put the bud into water, I shall tomorrow get a very different picture of my object. If I watch the rosebud without interruption, I shall see today's state change continuously into tomorrow's through an infinite number of intermediate stages. The picture which presents itself to me at any one moment is only a chance cross-section of an object which is in a continual process of development. If I do not put the bud into water, a whole series of states which lay as possibilities within the bud will not develop. Similarly I may be prevented tomorrow from observing the blossom further, and will thereby have an incomplete picture of it.
In the quote above, Steiner is speaking directly of our thinking activity's involvement in the rosebud phenomena we are observing, but here we will also use the rosebud as a symbol of our essential Thinking activtiy. If we were to stop contemplating that Thinking activity when we get to the form of mathematical thinking which now permeates modern science, it is as if we have failed to put the rosebud into water and/or failed to continue observing it altogether. We are then completely missing an entire panorama of cognitive modes which lay as possibilities within the seed of our essential Thinking. The mathematical thinking stage of each individual, and of humanity at large, is only a chance cross-section of thinking activity which is in a continual process of development; a continual "circumambulation of the Self". Unfortunately, ceasing observation of thinking is exactly what humanity has done in recent decades. Even the so-called "Darwinian" thinkers, who pride themselves on having a holistic processual understanding of life's history, fail to take up this processual understanding of their own cognition and that of humanity in general. That is something we should keep in mind as we wrap up our consideration of what our "I" knows in this essay, especially when we encounter the inevitable doubts. Most of those doubts, if we honestly reflect on them, will turn out to be the expression of our own incomplete observation of thinking, rather than any flaw in the logic itself.
To really understand what the development of mathematical thinking signifies, we will consider the relationship of outer sense-perceptions to inner mathematical concepts. First, let's provide some more historical context for this progression of 'knowing', which will also help us understand some key aspects of what actually occurred through it. In general, most people can sense that mathematical thinking, to the extent we have learned it and engaged with it, provides us with an inward certainty about the phenomena we are inquiring into; a certainty which non-mathematical observation and thinking do not provide whatsoever. For the latter, there is always a residual element of uncertainty and that does not sit well with the modern human soul. This soul, as we also observed in Part B, desperately wants to know about the world in excruciating, yet illuminating detail, and that is why it intently observes the world and thinks about what it observes. It was this 'certain' quality of mathematical thinking that was discerned around the late Middle Ages in Europe. This quality then motivated its adoption in all manner of inquiries, including religious, philosophical, and scientific ones, mostly in that order. To be clear, I am not suggesting that any particular individuals were responsible for bringing about mathematical thinking, but only taking note of how that thinking was reflected through their personalities.
Here we have expressed in the medieval era what was also prefigured by Socrates 1500+ years before, which we considered in Part A. It is the sentiment that we don't yet know a single thing about the essence of our experience and it is Wise to recognize that in humility. Eckhart could not sense any true knowledge of the Divine from the methods of knowing avaialble in his time. Therefore, he looked within, into the deepest "no-thing" of formless existence, and there he found the certainty of the "I" (Self) who is not other than the most high Divine. It is in this manner he felt confident in decalaring, "God and I are one in the act of knowing".
Cusa's quote comes from his key philosophical dissertation, Docta Ignorantia ("On Learned Ignorance"), and specifically Chapter 2 in Book I, "How it is that knowing is not-knowing". We immediately see the connection with Socrates' Wisdom. Cusa, however, was the first to go further and intimate that there may exist a faculty of knowing which can transcend mere rational intellect, which he referred to as "not-knowing". It is the sort of faculty which only dvelops once man learns, through knowing, of what precisely he is still ignorant - "For a man... will attain unto nothing more perfect than to be found to be most learned in the ignorance which is distinctively his". The rest of the dissertation goes on to speak of mathematical thinking as the means for the reasoning intellect to approach Divine truth, even if the highest of those truths remained just out of reach.
Could there be a greater contrast between Cusa's understanding of mathematical thinking and that of Spinoza? The 200 hundred year gap between them was more than enough time for the harmony between mathematical thinking and Divine truth, which had existed all the way back to Pythagoras, to be severed completely. The "divine truth" then was declared little more than "superstition" which could not a hold a candle to the mathematical "standard of verity". We are not passing judgment on any of the approaches or ideas here, only noting the major transformation in thinking about the phenomenal relations which took place. We could say man's reliance on Divine revelation for what he felt was genuine knowledge, born from within, officially ended with Spinoza.
After these three thinkers above, we see mathematical thinking featured heavily in the philosophy of personalities such as Leibniz and the science of Newton. The latter, in his famous Philosophiae Naturalis Principia Mathematica, stated as follows at the outset: "Hitherto I have laid down the definitions of such words as are less known, and explained the sense in which I would have them to be under stood in the following discourse. I do not define time, space, place and motion, as being well known to all. Only I must observe, that the vulgar conceive those quantities under no other notions but from the relation they bear to sensible objects. And thence arise certain prejudices, for the removing of which, it will be convenient to distinguish them into absolute and relative, true and apparent, mathematical and common." That Newton now takes time, space, and motion as "absolute" and "true" mathematical measures, regardless of one's perspective on them, really makes clear how far the mathematical thinking had progressed away from the qualities of experience at that stage. One's "point of view" was considered completely irrelevant to these pure mathematical concepts and, moreover, a source of "prejudices" which needed to be addressed. It was not until Einstein's observations in the 20th century, some 350 odd years later, that this "prejudiced" relational understanding was shown to be inseperable from the 'fabric' of space-time.
Remaining true to our phenomenological approach, let's return from that brief detour to see what we can discern about the relationship of mathematical concepts to the pheneomenal sense-world by considering only the givens of our concrete experience. We will take two 'objects' which seem completely unlike one another in appearance - (1) the "Moon" (outer sense-perception) and (2) the form of "triangle" (inner mathematical concept). Since we are picturing the "triangle" to ourselves, this mathematical concept has also become an inner percept. With these two objects, we are looking for what common element can be discerned between them without any added assumptions, i.e. without speculation about what "might be" similar between them. If we reason through the possibilities carefully, then we will see that no aspects of their outer appearances can fall into this category of "unquestionable common element". Any "triangular forms" we feel could also be present within the percept of "Moon" are just speculations without any way of verification. In addition, any triangular shapes we discern in the Moon-percept are not at all pure triangle-forms of the sort we are perceiving inwardly. So what else remains as a common element that we can discern between them?
Logic will eventually dictate that the only common element is the qualitative meaning. They do not both have the same meaning, but they both exhibit meaning in general. We can sense meaning in the Moon and all pure triangle-forms in our perception of both. The fact that this common element is so simple does not make it any less true. It is very simple but it is not at all obvious to the average person. As mentioned in Part B, we should expect Thinking which has factored its own thinking into phenomenal inquiry, thereby increasing its own depth and degrees of freedom, to render complicated phenomena more simple to understand. Although basic percepts of the Moon and triangles do not seem very complex, logic also dictates that meaning will be the only unquestionable common element between any set of outer sense-perceptions and inner mathematical percepts-concepts. So now we move on to explore the significance of this common element. What is the meaning of 'meaning' as common element, i.e. how does it illuminate the nature of modern scientific thinking? We should remember here what mathematical thinking takes away from the phenomena when it is developed inwardly as pure concepts. What is taken away is precisely the qualities of those phenomena - colors, sounds, tastes, smells, textures, etc.
Mathematical thinking takes one set of meanings away from the phenomena it brings within us - the qualtitative sense-meanings - and substitues another set of meanings, which is their quantitative measurements. The latter are not "measurements" that we have gone out with a ruler to determine in our inquiry, but rather in the sense of relational concepts - existing entirely within us as ideal forms - which allow us to more precisely compare phenomena to each other, and to compare the results of our mathematical comparisons with the results of someone else's comparisons. Take notice that everything related to this relatively recent mathematical thinking of humanity is about adding depth to our own thinking activity - it causes us to search for the meanings of meanings, to make comparisons of comparisons, to think, by way of our inner math concepts, about our thinking, which is first motivated in us by way of sense-observation. As we discussed in Parts A and B, this "self-deepening" quality of Thinking is absolutely critical for any true knowing endeavor. It is how we come to perceive the reasons why we can know what we know. Those reasons then allow us to take a higher perspective on our knowledge and evaluate what aspects of it are incomplete.
It is my argument here that, when we follow this logic carefully as it naturally unfolds, we will find most of our knowledge is drastically incomplete. The reason it is incomplete is rooted in the fact that the phenomena, after inwardly treated by mathematical thinking, is devoid of the qualitative meanings. How can one say scientific knowledge of the phenomenal relations is complete without any account for the most immanent qualities which shine forth from them? Goethe was the first modern thinker to realize that mathematical thinking, even when it comes to the physical sense-impressions and their relations, is only half-complete. He remarked in one of his Aphorisms in Prose, “Mathematics, like dialectics, is an organ of the inner, higher sense; its practice is an art, like oratory. For both, nothing is of value except the form; the content is a matter of indifference to them. It is all the same to them whether mathematics is calculating in pennies or dollars or whether rhetoric is defending something true or false.” So how is mathematical thinking rejoined with the meaningful content of the phenomena so that our scientific knowing is no longer indifferent to the Good, Beautiful, and True qualities of the world?
Since we have seen that meaning is the common element of all phenomena we perceive and think about, we must look to expand this meaning in a systematic, mathematical way. Consider what every scientist already does in their respective fields - they take observations and mathematical analysis of phenomena and derive "laws of nature" and "mathematical laws" which they hold as the underlying 'universal' principles explaining all of the particular manifestations of those phenomena. These underlying principles are, or should be, considered more real than the particular manfestations they are purporting to explain - the manifestations come and go and are entirely dependent on the principles, while the latter remain constant (albeit incomplete) and are entirely independent of the particular manifestations. Therefore, we should also expect the laws and principles of nature and mathematics, to the extent they have been reasoned properly, are more complete reflections of the meanings than those which arise from particular phenomena. However, in the case of mathematical concepts in most fields, the inverse (or antithesis) holds true. We can generally discern more qualitative meaning from the more simple concepts. Let's take a look at a few of those below:
"Point" - a particular spot, place, or position in an area or on a map, object, or surface:
"Line" - a curve connecting all points having a specified common property.
"Angle" - the space (usually measured in degrees) between two intersecting lines or surfaces at or close to the point where they meet.
The genuinely phenomenological approach here would be to dwell deeply with these meanings above and see what other qualities of meaning they reveal inwardly in our contemplative thought. We can imagine it as a process of broadening out the above meanings so they start to resemble something more like the natural "laws" and "principles" of meaning. All of those qualities taken away by pure mathematical thinking can then be restored back to these interconnected, interwoven constellations of meaning. Anyone who wishes to do so can take a few minutes in solace to do this exercise and see what results precipitate from the images. For our purposes here, we are going to take a slight 'shortcut' to arrive at those inwardly-sensed, more comprehensive meanings. All of these mathematical objects - point, line, and angle - are now also words which are frequently employed in what we refer to as "metaphors" and "analogies". These 'symbolic' usages are nearly always employed to direct the hearer or reader to an inward mental quality, state, disposition, or activity. Although philologists have been made a lot of progress at discerning exactly how these uses of mathematical words came about, that is not our concern here. Rather, we will see what their more "metaphorical" definitions reveal to us about their inward meaning.
"Point" - direct someone's attention toward something; direct or aim (something) at someone or something; an argument or idea put forward by a person in discussion; advantage or purpose that can be gained from doing something:
"Line" - a manner of doing or thinking about something; an agreed-upon approach; a policy.
"Angle" - a particular way of approaching or considering an issue or problem; a bias or point of view.
One thing we notice immediately is that all such metaphorical usages relate to intention and purpose (aim). That is, in general, what our ideas do - they contextualize the meaningful qualities of experience by structuring them within specific aims. When we have the idea of "going to the grocery store", all meaning that we encounter along the way is structured by this overarching idea. Yet "going to the grocery store" is also a concrete meaning structured within higher-order ideas, such as "things to get done today". That idea, in turn, is structured within "day of the week", "time of the month", "season of the year", so on and so forth. We can see how quickly these overarching ideas broaden out to encompass more and more constellations of meaning across space and time - the "self-encompassing" quality of Thinking. We should inwardly feel how the qualities which were taken away by mathematical thinking begin to return as we engage in this sort of qualitative inquiry. Colors, smells, sounds, tastes, textures; even shapes, motions, and transformations - these convey to us the meaning of life. Just as the percepts in the world do not appear to have any life without these qualities, the concepts also remain as dead husks of meaning until they are brought new life through our own Thinking activity. That is the "self-vitalizing" quality of Thinking. Let's further contemplate the common constellation of meaning of these mathematical concepts which revolves around a way of seeing the world; around the "point of view" which Newton deemed irrelevant to "true" space and time.
Nature is made to conspire with spirit [Thinking] to emancipate us. Certain mechanical changes, a small alteration in our local position apprizes us of a dualism. We are strangely affected by seeing the shore from a moving ship, from a balloon, or through the tints of an unusual sky. The least change in our point of view, gives the whole world a pictorial air. A man who seldom rides, needs only to get into a coach and traverse his own town, to turn the street into a puppet-show. The men, the women,—talking, running, bartering, fighting,—the earnest mechanic, the lounger, the beggar, the boys, the dogs, are unrealized at once, or, at least, wholly detached from all relation to the observer, and seen as apparent, not substantial beings. What new thoughts are suggested by seeing a face of country quite familiar, in the rapid movement of the railroad car! Nay, the most wonted objects, (make a very slight change in the point of vision), please us most. In a camera obscura, the butcher's cart, and the figure of one of our own family amuse us. So a portrait of a well-known face gratifies us. Turn the eyes upside down, by looking at the landscape through your legs, and how agreeable is the picture, though you have seen it any time these twenty years! In these cases, by mechanical means, is suggested the difference between the observer and the spectacle,— between man and nature. Hence arises a pleasure mixed with awe; I may say, a low degree of the sublime is felt from the fact, probably, that man is hereby apprized, that, whilst the world is a spectacle, something in himself is stable.
- Ralph Waldo Emerson, Nature (1836)
It is nearly impossible, after sufficient reflection, to deny that our view of the world has been mechanized by mathematical thinking and the technology it has made possible. Just as the qualities were stripped from the sense-perceptions by the inward mathematical concepts, the world around us then came to resemble one big and dreary mathematical concept. Nearly 200 years after Emerson wrote the above, in the 21st century of digital technology, that fact has become even more true. The temporal gap between the deadening of our conceptual thinking and the deadning of the phenomenal world, via our technology made possible by that thinking, has been nearly eliminated. It would really seem as a hopeless affair if history and experience had not already shown, beyond a shadow of a doubt, that the world does not stop evolving in our "present" moment. Even within mathematical thinking, at a broader temporal perspective, we can see how it evolved in a sort of "V" shape in the modern age - first developed abstract algebra and geometry, which focus on segmenting phenomenal relations into discrete information-packets to analyze them separately, and later we get the calculus of Newton and Leibniz with its "differential" and "integral" equations, which focus on putting back together what earlier mathematical operations had taken apart.
Still, we must remember that the self-deepening quality of Thinking makes the phenomenal relations easier to understand, not harder. The reason why the equations of calculus wash over most as if they are indecipherable alien languages is precisely because they have no living qualities we can relate to anymore. When a person's thinking does not appear to have any such qualities, we aptly say, "you have no Imagination". That is really the quintessential case of a person living in a glass house and throwing stones. "Let he who is without Imagination cast the first stone". What Coleridge referred to as the "philosophic organ" at the beginning of this essay is this Imaginative thinking-sense. It is also what Goethe was pointing us towards when he remarked, "The phenomenon must never be thought of as finished or complete... but rather as evolving, growing, and in many ways as something yet to be determined... ‘Vernunft' [Reason] is concerned with what is becoming, 'Verstand' [rationality] with what has already become … [Reason] rejoices in whatever evolves; [rationality] wants to hold everything still, so that it can utilise it'." What Goethe remarks about the evolving "phenomenon" must be applied to our own Thinking activity as well. As we have seen, that activity is inseperably bound up with the phenomenal world we perceive.
To deny the possibility of a precise, systematic, 'mathematical', and Imaginative mode of Thinking is like a purely instinctive animal (if it could somehow think and speak) denying the possibility of Reason and Imagination which has evolved from its own instincts. All of what we have discussed here flows naurally from the very nature of Thinking itself - all of its "Self-revealing" qualities - as we find it expressed in our immanent experience of the world. It is this deep, reflective consideration of our Thinking activity which then logically neccessitates the evolutionary path we have charted out. The modern German idealist philosopher Hegel spoke of this evolution as a dynamic between thesis (investigation of phenomena by outward sense-observation and basic thinking) and antithesis (empirical science by way of inward mathematical conceptual systems), which results in synthesis (rigorous empirical science by way of Imaginative thinking which transcends outward-inward distinctions). Every stage in this evolutionary process contains all previous stages. When the Whole is carved up and viewed from its various parts, we call that philosophy and science. When those knowing inquiries yield back to us the Whole from a higher vantage point, it is called true mathematical and imaginative knowledge. And, when we consider the entire holistic process and know that only it, through us, can bring about our soul's deepest satisfaction, we are imagining Faith and Wisdom.