#numerical systems

This is an extract from a huge conversation I've had with a good friend (Simon Madsen) following the Gas video post 'Microscopic'

BR: It's uncanny that 10 to the power of 1,2,3,4,5,6,7 brings us exactly to the outside radius of our planet. Like someone planned it or something...

SM: It puts numbers (and us) into perspective. But 10 to the power of 7 is 10,000 kilometers. The actual diameter of Earth is closer to 12,500 km (which is the mean diameter since Earth is not actually a sphere) so it's not really that uncanny. And someone did plan it: humans, who invented numbers.  But it's amazing how numbers can "cheat" perception by adding zeros: to our mind, it's not really that far from 10^7 to 10^42, relatively speaking. But one is about the size of earth. The other is the length of the known universe. Yes, 42. Now that's uncanny.

BR: Proportion is something that was inspired by naturally occurring patterns such as fractals (otherwise we would have no basis to develop a numerical system on), so it's a bit of a chicken and egg argument. But yes, it's commonly understood that 42 is the the answer to life, the universe and everything. That and a nice hot cup of tea.

SM: I don't know if that is true. Numerical systems are usually attributed to India (for counting) and Egypt (for geometry), right? But at that time (5000 to 7000 years ago) no one knew about fractals and the rest of complexity theory. Numerical systems were based on their relevance to human affairs: commercing and building houses - in my understanding. It doesn't take knowledge of fractals to count to 9 - and as Richard Feynman said, math is nothing but counting and deducing shortcut rules for counting.

Only when logic and math were disembedded from practical use did the underlying connection between numbers and nature become clear - like fractals, pi or the Fibonacci sequence. So no, I didn't mean that humans invented 10^7 as being the relative diameter of Earth, but without our need to quantify things - and later concepts - these "magic coincidences" wouldn't have occured. So to keep your chicken-and-egg argument, I believe the chicken came first (the chick being numbers, the egg being magic coincidences like fractal phenomena).

BR: The origins of mathematics in terms of the role it played in commerce and exchange are pretty well understood, but the psychological development of arithmetic (or why a numerical system came about at all) was more what I was getting at. In my view, it was shaped by the natural phenomena we are surrounded with. Just because it took us several thousand years to determine the link between nature and mathematics does not mean that the proportion of naturally occurring objects haven't influenced us through-out our existence. It is recognised that cyclical phenomena such as day and night or tidal shifts are instrumental in our perception of time, so it was with similar events in our interpretation of arithmetics, it couldn't have been otherwise.

You can argue that the significance of the number ten or 'adding a zero onto the end of things' is purely etymological, you find that the pattern that this reflects is mirrored in nature, the radius of the Earth at approx. 10^7, like the koch snowflake or Dragon curve or the examples you mentioned such as the Fibonacci numbers. Just because it is a construct we have discovered or interpreted does not make it abstract from nature, nor does it make it a purely human interpretation. It's a system that is potentially billions of years old, but one that we have come to interpret on a basic level and one we occasionally find patterns in. When we find a pattern, we tend to shake our fist at the sky or blow ourselves up in spectacular fashion. Yet, the empiricist in me can't help but think that it wasn't purely a human construct, but simply an realisation of a system we didn't create and are just beginning to understand. We have learned to recognise patterns in it, but certainly are not the creators of it. To make the point, the chick you described wasn't born yet when the egg you described hatched!

Far from saying it's magic, it is indicative of a system learned by phenomena we have been surrounded with since our existence, and then (in the last 400 years) we have discovered that the application of these basic systems can be advanced to encompass a macro scale (we chanced upon by studying phenomena) as well as on the micro-scale we have been aware of all these thousands of years such as counting the number of cows we own. This for me is one of the more amazing concepts brought to life in that video.

SM: "I suggested it was as though someone had planned it."

And I correlated this suggestion with magical ones, i.e. that their is some unknown entity or system of meaning that humans are in the process of unraveling. This is the position I'm speaking against. Magic might not have been the best choice of term. 42 might be better: there is no number 42. We ascribe to it it's meaning as the answer to everything.

The key point I'm trying to make which, as I see it, stands in opposition to yours is that there is no meaning inherent in the universe for us to discover. No meaningful relationship between numbers, or anything else, unless we at some point decided that there should be, consciously or not. All meaning is imposed through language. This might be a radical and ultimately flawed position to maintain, and I'm admitting that this is a far, postmodernist reach which I might not be able to uphold or even completely agree with, but for arguments sake I'm going to continue fearlessly out to where the ice begins to crack.

"In my view, [arithmetic] was shaped by the natural phenomena we are surrounded with."

Well, there is no argument here. This is an arbitrary point since we are all of nature and shaped by nature. It equals, logically, to saying that 'some of everything' is shaped by 'everything' or that all modes of thinking is in some way shaped by the context in which 'it thinks'. It's a given. Like, it's a given that 1 cow will, at all times by all counting creatures in all universes, always sum up to just 1, provided there is agreement on what constitutes '1' and 'cow'. I regard this as our common presupposition. 

What is not our common presupposition, though, is that language in your positivist/empiricist view is regarded as transparent, or directly mediating the ontology of the universe between nature and consciousness. This is the presupposition I'm speaking against.

So when we count 1 cow, we are immediately attaching the socially negotiated concept of 'the number 1' to the shape we, through similar processes, call 'cow'. Without these terms, no meaningful relation can be discovered, extracted or invented. Nevertheless, I agree that "It's a system that is potentially billions of years old, but one that we have come to interpret on a basic level and one we occasionally find patterns in." But the patterns we find are not embedded in the universe for us to eventually and necessarily discover. They are contingent claims of meaning correlated between objects as they are revealed to our senses and categorized in accordance with the dominant linguistic discourse. Hence my disagreement with your saying that, "we have learned to recognise patterns in it, but certainly are not the creators of it." 

So my reproduction of that statement would be: We have learned ourselves to recognize patterns in it, exactly because we are the creators of it.

"Just because it took us several thousand years to determine the link between nature and mathematics does not mean that the proportion of naturally occurring objects haven't influenced us through-out our existence."

True in the sense that the world existed before we humans came about. I understand this as a rebuttal of my saying that inventing numbers 'came before' discovering, say, the relation between a circle's circumference and it's ratio. because, as I assume you maintain, the number pi was their to be discovered all along.

Here, a distinction need to be made between ontological and epistemological claims: yours is of the former kind, mine is of the latter kind, since I cling to the premise of there being nothing outside of epistemological claims, through which all of ontology is understood and ascribed it's meaning. So to say that fractal phenomena, for instance, existed way before humans could make sense of them is therefore a meaningless statement. In other words, for something to be intelligible, it has to be attached to a set of linguistic concepts or pictures. Again, the meanings imposed on the shapes of the universe are arbitrary and socially negotiated. You can say that pi is a naturally occurring number or relation between objects that we discovered, but by stating this you adhere to the anthropomorphic presupposition that human consciousness is somehow fundamental to the ontology of the universe. This seems to me dogmatic.

(There is certainly strong reasons to believe that nature often follows a fractal structural design, but it is impossible for us to determine if this is the 'true' structure of the universe, since we must constantly refer to preexisting concepts in our description of it. New words like 'fractal' are thus constructed from the meaning attached to old words, and there is then no ontological certainty as to where one fractal ends and another begins; only our perceptual experience of a vague resemblance between clouds and coastal lines.) 

"We have discovered that the application of these basic systems can be advanced to encompass a macro scale (we chanced upon by studying phenomena) as well as on the micro-scale we have been aware of all these thousands of years such as counting the number of cows we own."

They only manage to encompass different scales when we shift from one discourse to another - what Thomas Kuhn described as 'the incommensurability of scientific paradigms'. Like the shift from Newtonian mechanics (which works on an Earthly scale, but not on a universal one) to Einstein's theories of relativity (which works on a universal macro-scale, but not on atomic micro-scale). And, as you know, we are yet to discover (read: invent) a system commensurable between micro and macro in order to fully explain the workings of gravity in these new physics.

Being the Devil's advocate, I will stress the provisional nature of this text with a quote from former Nature editor in chief, Philip Ball:

"From here it is but a short step to Pythagorean mysticism. But that need need not distract us from responding with a certain wonder at the universality that organizes many aspects of society in the same way that it directs the properties of atoms. We need not turn this into a 'religion of science', any more than we need regard the whorls of a flowing stream as evidence of divine planning. We can simply celebrate the fact that their are indeed 'laws of large numbers', and that they let us divide order and regularity in an otherwise terrifying diversity."

My emphasis on that last sentence. The universe moves about itself in a certain way rather than in other ways. We call that way a 'constant' and derive rules and laws from it by induction and deduction. But the medium by which this takes place is influencing our results in another 'way'. I find it incredibly hard to separate one way from the other.

BR: I believe that science and mathematics are able provide an understanding of formerly inexplicable phenomena, to provide context and probably most reassuringly for the fragile human psyche - give hope that currently inexplicable phenomena might one day be explained. I believe in 'cosmic indifference' and do not think there is an inherent fairness in things.

When we observe patterns in nature we are of course observing this within the context of own subjectivity, but we are capable of using this information to create connections and observations that allow us to make sense of what we are experiencing. In some cases control what we are experiencing and further our understanding (and curiosity) to better our rationalisation and interpretation. In terms of understanding, mediating the language of the universe to our consciousness, this is a much more substantive debate. I believe we are furthering our knowledge of the phenomena that surrounds us and utilising this in a way that furthers our understanding to the point of which we determine the ultimate cause of it, to know in a sense how it got here. I do not think this is the universe or the constant conversing with us giving leaving us clues in an Arthur C Clarke-AI sense, but it is out consciousness coming to terms with seemingly random, terrifying phenomena and developing the broad awareness of systemic understanding. Systems are something that have come inherently from our consciousness and as a species we use to break complex data down into understandable chunks (as do most conscious animals), but as we broadly become more intelligent and less terrified, our ability to interpret complex data increases. The more complex the data we break down the further our understanding goes and from here the capacity to take in more and more stimuli produced by the wider universe increases to the point where we identify systems, patterns and correctly predict events that our ancestors believed were acts of god or meaningless nonsense - this is disassociated with linguistics and takes us to a higher plain of understanding that usurps the process of societal constraint (but not subjective of psychological constraint). In this instance, I disagree with your point: "We have learned ourselves to recognize patterns in it, exactly because we are the creators of it." 

 To the contrary, we have learned to recognise patterns in it because there are patterns in it. If there were not we would not recognise them, nor would we be able to predict the re-occurrence of phenomena that is outside of our design or control. We know this because we are very good at determining when there is no correlation or causality, humans have words such as 'fate' or 'random' or 'luck' to interpret events that sit outside of our understanding and do not fit a set pattern. You say that meanings imposed on the shapes of the universe are arbitrary and socially negotiated, "everything given meaning by our own interpretations" does not allow for when patterns that are our own construct are found to be repeated outside of the context of their discovery and in ways that don't fit into the interpretative binds we have created for phenomena. To suggest it is all random chaos and that any system of order you find in chaos is a fallacy - a subjective construct - falls wide of the mark when forms of 'order' reappear outside of the pre-subscribed (and very limited) models we have developed. I believe we are etching the first basic cave-paintings of scientific understanding for a much more complex and all-encompassing system that exists. I agree that it is hard to tell the difference between inexplicable phenomena and systems, we are often very good at prescribing meaning to things that have none, such as the example you give of clouds resembling coastlines. But I don't think we should call off the search for meaning and a systemic function inherent in things simply because we get it wrong from time to time. Nor do I see the position of assuming everything to be chaos because we are as of yet to make sense of all the stimuli, data, phenomena (call it what you will) that exists. We have come a long way from being overwhelmingly terrified by everything to becoming profoundly curious about everything in a few short millennia, already a leap for any conscious species. But I agree with Kuhn, Ball and yourself that we are very far from being able to apply our understanding about the universe to it in any meaningful way, our science I think is at the level of cave-paintings of our ancestors to the presumed roof of the Sistine Chapel that stands before us - namely very low in our understanding. I don't doubt that we have misinterpreted and misunderstood, but I also do not think an interpretation and an understanding is beyond our reach.

SM: I certainly agree with your description of scientific realism. It has given us 'true meaning' in both practical application (mostly) and some rare theoretical insights like time relativity and thermodynamics. Personally, I too consider reality to be there, knowledge to be growing and energy and/or information to be dissipating. I follow your argument about the scientific method allowing us to "see past" socially constructed norms, from observed orderliness to theory to verification/falsification of theory outside of the original context. Constructionists typically rival this argument with the theory-ladenness of science or, as I have done above, with the psychological dependance on preexisting concepts for new descriptions. Nevertheless, the scientific method stands it's ground, still, primarily owing this position as the perceived pinnacle of human invention to it's continuing succes in practical applicability.

My current position as the naive relativist only hinges on the concept of meaning - or the 'making sense of something'. So I will attempt the argument for scientific constructionism through the concept of truth.

It is fair to say that through science humans have gained understanding of themselves and their environment. Knowledge, it is said, accumulates in culture in much the same way gene information accumulates in nature: as a response to stimuli organisms get better, know better. A constructionist would agree. For there to be stimuli there must be some things that stimulate. And for there to be stimuli there must also be some things affected. For there to be consciousness there must be some things that experience the stimuli. In fact, all that consciousness can ever experience is stimuli. There is no experiencing the thing that stimulates, the thing-in-itself, only the vividness of the experience itself, the thing-for-us (the Kantian contribution). Hence our subjective 'entrapment', which you pointed out. 

The scientific method wrestles itself loose from this hermeneutic circlejerk by quantitative approach: we know we are all 'in here' and can never get out. But if we record enough instances of something in experiment A, and this record is reproduced somewhere else (out of context) in experiment B, we must assume that reality in this instant is equal to both A and B (and C and D and so forth), and we produce a scientific theory of universal consequence. Some years later we might discover a new relation, in a completely unrelated field, that seems to also corroborate this theory. So the theory is verified as true. And the theories build on top of each other and it gets ever more easy (but not really) to see when new suggestions doesn't fit in with what is now an established framework or a 'worldview'. Details here and there get corrected, occasionally a whole theory is overthrown by a new and more accurate one, but the framework itself is never totally reevaluted. Incidents and discoveries are predicted within the framework's 'blank spots', and they are later verified by experiment. It's nice and tidy.

But nowhere in this process, I submit, has the theory escaped social interaction - the working together that makes science a communal discipline. The framework is still true, though only to us. What relationships and patterns we find are only intelligible to us if they fit inside the framework (they are theory-laden). When we say that something is true, we presuppose the thing-for-us vs. thing-in-itself problem. We reduce the concept of truth to count only for the thing-for-us, since we know we can never experience the thing-in-itself (though we agree that it exists independent and indifferently to us). This reduction of the meaning of truth has the logical consequence, that it becomes a question of human experience in society whether something is true or not. Since the framework is built on arbitrary concepts, for which connotations are slowly drifting according to discourse (but never steady), it can never speak directly of the 'nature' of reality apart from human existence. So even though patterns may be there in reality for us to stumble upon, we can never shake the inborn interdependance of a pattern's perceived orderliness and our subjective viewing of it. It goes together - always. So to whom belongs the order? That is answered when you answer to whom it makes sense.

But why care if its only relevant to humans? That's what we are and that's how it is. It has worked so far (not counting minor hick-ups like purgatory and alchemy) so we should just keep going, right? 

There might not be a problem if science never encountered major insoluble problems. When our successful theories and powerful automata brought our experience of the universe down to the smallest conceivable notion of matter and force such a problem emerged. In quantum mechanics our hitherto successful chain of causality broke. The very act of observing something made it unobservable (it's a wave...no it's a particle...no wait, it's a wave! Well, it is really a particle behaving weirdly). So we experienced the limit of our experience, so to speak. There are no more 'if-then' predictions to be made and no more types of reality to be observed and no more scales of smallness to be discovered (in this field). Theoretically we are stille blasting away at full speed adding dimensions to this and that, but the practical falsification of a theory of the make-up of reality has stopped. Not because we got to the end or need to build bigger magnifying glasses, but because our very existence is so deeply embedded in the reality we occupy, in our experience, that the fact of existence itself disturbs our ability to see further. This resonates with me in a profound way. It is then that I start to believe that our incessant interrogation of our surroundings is drenched with assumptions of our own making to the point where it is nearly impossible for us to tell the difference. This is the moderate constructionist position.

But I do not submit that 'everything is just chaos' then. Even in our caveman-like understanding it is rather intuitive that some forms of orderliness are preserved along with the chaotic. The fact that we have these experiences at all testaments to orderliness of a rather complex kind. But our description of it will always depend on the experience of the thing-for-us. And in describing it, we will make use only of floating signifiers that already make sense within the framework. And our understanding of it will never leave the skull of it. 

I hear you say that time and shrewdness will give us a complete, objectively true theory of everything. And I cannot see how this can happen under the the ever-present psychological and social constraints of the human condition. 

Also, through all the breakthroughs made in the natural sciences - and in part in philosophy - the goings on of satisfying our scientific curiosity has only very recently begun to launch fully armed investigations into language and social practice. If the natural sciences are but cave-paintings the social sciences are still being chucked from flint stone. I believe revolutions are to come in the understanding of the role of social interaction and communication that will shake some all to well established scientific foundations.

I'm freely translating my favorite Niels Bohr quote, because I can't find it in English:

"It is in the description of Nature not suggested that we uncover the proper essence of phenomena but only in the widest possible sense trace correlations in the multiplicity of our experiences." 

I still love all things science, though :) But I believe that science, like religion and art, assumes, wrongly, some sort of human superiority and at the same time tries to gratify a primal yearning to return to a state before complete self-awareness. As anthropologist Wade Davies says of his friend and fellow scientist, Clayton Eckleman:

“Clearly, at some point we were all of an animal nature. And at some point we were not. Proto-shamanism as an original attempt, through ritual, to rekindle a connection that had been irrevocably lost."