Responding to Ed George About Mathematics

_{William J Murray
December 13, 2018

Intelligent Design

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Intelligent Design}

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In another thread, Ed George insists that humans invented mathematics as a way to describe the behavior of phenomena, but that doesn’t mean mathematics is an intrinsic aspect of the universe, a part we discovered, not invented. Here’s why that position is untenable.

Mr. George is correct that humans invent languages – the language of mathematics included. Languages are systems of symbols that represent things. For example, the word “sphere” can be expressed with different symbols in different languages, but the symbols all refer to the same thing – in this case, the form of an object in the real world. That we invented the symbols and language to describe a real thing doesn’t mean we invented the real thing itself.

As Mr. George agrees, mathematics (in terms of this debate) is an invented system of symbols used to describe behaviors of phenomena (physics).

However, humans did not invent those behaviors; we are only describing them using symbolic language. Phenomena in the universe behave in, let’s say, “X” manner. X is a set of discoverable patterns. We discovered those patterns and applied symbolic language to represent and calculate them. In the same way that “sphere-ness” is an inherent quality of something in the universe which we use the term “sphere” to represent, “mathematics” is a term we use to represent an inherent quality of the universe.

Yet, Mr. George denies that we can know whether or not we “discovered” these behaviors (which we call “mathematics”. Of course we did, and we use symbolic language to describe those qualities and behaviors we have discovered.

This same, simple logic can be applied more broadly. We invented a symbolic language in order to refer to things we discover about our existence and the universe, as KF is pointing out, in terms of logical first principles. We did not invent that 1+2=3; those symbols represent observable facts. We did not invent the principle of identity out of whole cloth; it represents an observable fact and, more deeply, a universal structure that human minds cannot escape, no matter how hard we try or imagine. As KF points out, it is responsible for our ability to have cognition at all or to invent and use language. Logical first principles are a fact of our existence which we discovered – first as “X”, then using a string of symbols to represent.

Beyond observable facts, such symbolic language can represent other discoverable facts; such as, some things are impossible to imagine. Imagine that 1+2=4 in any observable way. You can say the words or write the equation, but it is not possible to imagine it being a discoverable fact in any scenario. It’s a nonsensical proposition, much like a 4-sided triangle. The inability to imagine a thing has other implications, but that’s for another conversation.

Language is the invention, but language is itself governed by certain necessary rules. Those rules were entirely hidden to us in the beginning, but we know they were there because inevitably all languages follow those fundamental rules even if we are unaware of them, the first of which is the principle of identity. Without that, language is impossible.

These “X” characteristics of our universe and our existence are things we discovered and then used symbolic systems to represent.

Comments

? So what is the chuckle about? We seem to agree about the value of complex numbers as ways to represent vectors, which is a very useful and powerful part of modern science.hazel_{December 16, 2018
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H, I chuckle. The point of course is study vs structure. Complex numbers, particularly in real + imaginary parts form, turned out to be manifestations of vectors in the plane and useful forms for expressing same. KFkairosfocus_{December 16, 2018
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Again, I'm quite familiar with everything in the Wikipedia post, FWIW. But it's fun to read it. And yes, vectors have many pure and applied uses: very important tools in the math toolbox.hazel_{December 16, 2018
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H, sometimes the idiosyncrasies of history can make an issue more complex than otherwise. In my experience of sneaking in complex numbers a year or two early in order to better address AC theory or the like [and yes I preferred Lorentz force to the hand rules too to explain motors and generators], the rotating vector approach did not produce the reactions that I recall from 6th form Mathematics. That was a lesson. Coming back to the focal issue, it seems clear that "complex" numbers are complex as they are vectors. Indeed by historical way of Quaternions, so are ijk vectors. The quantitative structure in view then is vectors in a certain abstract logic world that captures key features that then slide over into ever so many other "pure" and "applied" contexts. KF PS: Notice Wikipedia's summary:
A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is a solution of the equation x^2 = -1. Because no real number satisfies this equation, i is called an imaginary number. For the complex number a + bi, a is called the real part, and b is called the imaginary part. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers, and are fundamental in many aspects of the scientific description of the natural world . . . . The 16th century Italian mathematician Gerolamo Cardano is credited with introducing complex numbers in his attempts to find solutions to cubic equations.[4] Geometrically, complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex number a + bi can be identified with the point (a, b) in the complex plane. A complex number whose real part is zero is said to be purely imaginary; the points for these numbers lie on the vertical axis of the complex plane. A complex number whose imaginary part is zero can be viewed as a real number; its point lies on the horizontal axis of the complex plane. Complex numbers can also be represented in polar form, which associates each complex number with its distance from the origin (its magnitude) and with a particular angle known as the argument of this complex number.
PPS: Rearranging that exponential vector expression to give 0 = 1 + e^i*pi then reveals a key, infinitely deep structural coherence across several domains of Mathematics. I think that is part of why it is rated the most beautiful equation in Mathematics. This expression stands in handily as poster child for the logic of structure and quantity. Tendencies to down-play it are in my view symptomatic.kairosfocus_{December 16, 2018
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Yes, kf, I am familiar with the history of complex numbers. The invention/discovery of the complex plane as a way of representing the complex numbers as vectors was a powerful addition to mathematics, just as the number line as a representation of the real numbers was. I taught the beginnings of all this to my pre-calc and calc students. For instance, as an exercise in learning the arithmetic of complex numbers, I would have students use the quadratic formula to find the complex roots of a quadratic such as x^2 +2x +5 = 0, and then substitute one of the solutions back into the equation to demonsrate that it was indeed a solution. They were impressed that it all worked, and proud of themselves at times for being able to do it (as this was all pretty new to them.) I also showed them that connecting the endpoints of the n nth roots of a number, when plotted on the complex plane, made a regular polygon of n sides, and as a bit of a challenge had them find the three cube roots of 1 and show that they all, when cubed, did equal 1. And, of course, we learned Euler's formula for e^(ix), did some problems with it, and learned why e^(i*pi) = =-1 is just a direct result of letting x = pi. All good stuff. One of my favorite topics.hazel_{December 16, 2018
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H, the structure captured by the use of complex numbers is 2-d vectors in a special plane; which is a much more conceptually tractable matter than imaginary roots. And BTW "complex" strongly suggests, vectors. This approach/ perspective allows us to identify that numbers suitably represented can be 2-d quantities bearing magnitude and direction from the origin and even as rotating in time, e.g. Z = R*e^i*wt which points to the power series structure too. What came up as a way to suggest solutions to all quadratic expressions and required positing square roots of negative numbers despite the fact that [-x]^2 --> x^2, led to a gateway into a powerful domain. KFkairosfocus_{December 16, 2018
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Hazel, many people, and I am embarrassed to admit that I am often one of them, don’t understand the passion that some people can have for mathematics. And then I realize that I have the same passion for analytical chemistry, something that relies heavily on physics and mathematics.Ed George_{December 16, 2018
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Thanks, Ed. I very much enjoyed making math concepts clear, interesting, and even exciting for my students. I taught everything from general math to 9th graders to calculus (for many years) to seniors, as well as College Algebra for a local community college. I especially concentrated when I could, and especially in tech math for the non-college bound, pre-calculus and calculus, on real-world applications.hazel_{December 16, 2018
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Hazel, I am really enjoying your comments. You present them with such clarity that you must have been a great teacher.Ed George_{December 16, 2018
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That is true: math constantly grows, and often unforeseen developments improve or expand a concept. It's all cool how it works.hazel_{December 16, 2018
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H, yes, the numbers were introduced for algebraic reasons. My point is that often a phenomenon is run into in an odd corner then somehow we blunder our way into a wider, better picture and see things more clearly. This happened with quantum theory for example. I suspect Newton's approach to fluxions is less fruitful than infinitesimals, and they went in abeyance in turn. Then non-standard Analysis came up. KFkairosfocus_{December 16, 2018
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Yes, kf, using complex numbers to represent vectors was a great invention/discovery. Visualizing complex numbers on the complex plane is one of the things I like about them best. But historically this came after just using complex numbers algebraically in equations. I think this is correct,hazel_{December 16, 2018
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LC, Sorry but a couple of emergencies just came in over the phone so I don't have time to respond on points to your interesting suggestions just now. Please see my remarks: https://uncommondescent.com/philosophy/logic-and-first-principles-5-the-mathemat-ical-ordering-of-reality/ KFkairosfocus_{December 16, 2018
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H, Pardon again, I know i was put on the table in a context which invited controversy -- indeed the sort of historical objections hinted at popped up right there in the sixth form classroom. What I pointed to was the vector phenomenon, which converts basic quantities into a plane of values. That vector approach then allows a far more natural interpretation of the exponential form and opens up the world of phasors, rotating vectors. This also points to the complex frequency domain. KFkairosfocus_{December 16, 2018
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I earn a living developing practical solutions to complex problems and I know my way around the mathematical block. I'm not an expert in theory. Math is a logical statement expressed in symbolic language. If the question is whether math is discovered or invented, you could ask the same thing by asking if logic itself is discovered or invented. The answer to one is the answer to the other; math can't be discovered if logic is just an invention. Math would be an invention too. The symbology and language used to express mathematical concepts is invented. If a given set of symbols is used in math, and the logic behind their use is identical to the logic of a different set of symbols, this does not make the math itself different. I don't see how anyone would disagree with me so far, but since free will does exist, I'm certain some will just because they can. And it's usually those that dispute free will, if that makes any sense. It has been claimed that since reptiles, birds and mammals all have been found to have a basic number sense and can readily see the difference between two and three, Human math ability is just built into us by evolution. So the idea that 1 + 2 = 3 does not have much of any implication as even lizards appear to know this. There is a wild Crow living near me that can count to 6 and there are accounts of other birds in other places with seemingly very canny ability to do addition and subtraction. I believe this means that math is not limited to just Human cognition. Indeed if math is invented it has been invented by several species. I find it amazing they all reach the same results. I think the best argument is that math, like water, food, shelter and other things is part of nature and all species take advantage of them if they can.LoneCycler_{December 16, 2018
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H, pardon but I am highlighting the subject AND substance question -- as I have emphasised throughout. On this particular issue Zeno's paradoxes of motion are an indicator of how the two must meet on equal terms. KFkairosfocus_{December 16, 2018
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math guy: Then how do we as purely material entities contact or perceive the platonic realm? Of course, this question asserts a fact-not-in-evidence as its premise: that we are "purely material entities." The answer to the question is "consciousness." A/mats assume that consciousness is a product of material processes. Others, such as myself, think consciousness transcends space-time and exists in the "platonic ontology." (Mathematics isn't the only realm where this sort of discussion exists. Musicians and music writers with a philosophical bent have these same sorts of dialogs. Some of us are music platonists some are not, and reasons are essentially the same. I, personally, think that all genuine creativity, contra synthetic recombination, comes from "there.")mike1962_{December 16, 2018
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re 112: it was originally an invention to just stand for sqrt(-1). The history of this is quite interesting, including the fact that many mathematicians rejected the concept for a long time. I believe lots of work with i as a number preceded the invention of vectors as a way of representing directed numbers.hazel_{December 16, 2018
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I know what calculus is, kf. :-) For many years, I enjoyed introducing beginning students to the concepts, techniques, and applications.hazel_{December 16, 2018
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F/N: Is "i" -- i = sqrt(-1) -- an invention or simply a notation for vectors in a particular plane of interest, the complex domain? KFkairosfocus_{December 16, 2018
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JAD & H: Calculus (more broadly, Analysis) is the study, rates and accumulations of flow, change, motion etc are substance and have always been with us as a quantitative, structural feature of a world where change can be discrete or continuous and variable. KFkairosfocus_{December 16, 2018
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Newton and Leibnitz both have some claim to having first developed calculus. Newton is usually given the most credit.hazel_{December 16, 2018
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I am simply underscoring the point that I made earlier @ #12 when I quoted Roger Penrose.
It is very important in understanding the physical world that our way of describing the physical world, certainly at its most precise, has to do with mathematics. There is no getting away from it. That mathematics has to have been there since the beginning of time. It has eternal existence. Timelessness really. It doesn’t have any location in space. It doesn’t have any location in time. Some people would take it not having a location with not having any existence at all. But it is hard to talk about science really without giving mathematics some kind of reality because that is how you describe your theories in terms of mathematical structures…
https://uncommondescent.com/intelligent-design/responding-to-ed-george-about-mathematics/#comment-669708 Who discovered (or invented) calculus?john_a_designer_{December 16, 2018
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PS: Lucasian chair, note Newton to Hawking. https://theconversation.com/from-newton-to-hawking-and-beyond-a-short-history-of-the-lucasian-chair-40967kairosfocus_{December 16, 2018
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H, the V as palm spread out looks pretty clear to me, though hash marks -- digits as unary numerals -- is suggestive and obviously can be seen as record of a finger. Digit of course pointing to finger. Tally sticks would be a natural extension. I forgot, back in 4th form Physics, definition no 1 was: the definition of a quantity or unit is a precise statement describing that quantity or unit. KFkairosfocus_{December 16, 2018
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JAD, at the time physics was not really separate from Mathematics, for obvious reasons. Newton would have termed himself a natural philosopher. The Lucasian Chair is currently deemed Mathematical but has been held by Physicists. Newton's opus magnum is the Mathematical Principles of Natural Philosophy. Yes indeed to envision the fields as fluxes evenly flowing from/to a pole immediately implies inverse square law dependence as a conservation of flux principle. KFkairosfocus_{December 16, 2018
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Pardon typo, 5.kairosfocus_{December 16, 2018
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One of the most significant discoveries in science was the discovery of the inverse square law (credited to Kepler for light) which is derived directly from the geometry of a sphere. The ISL applies to both electromagnetism and gravity, though the force constants for each vary. https://www.thehighersidechatsplus.com/forums/media/inverse-square-law-and-wave-function.105/full?d=1503980290 Where would physics be without this discovery? Contrary to popular accounts Kepler and Galileo were not astronomers and Newton was not a physicist. They were all mathematicians who believed that at its root the universe was mathematical.john_a_designer_{December 16, 2018
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Hmmm. 5^2 - 1 = 24, which is a multiple of 24, so why wouldn't one say "every prime from 4 on [not 6] is such that on squaring will be one more than a multiple of 24"?hazel_{December 16, 2018
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re 100: That is a neat fact, and a nice proof.hazel_{December 16, 2018
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