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

Hi math guy. First, I am a fan of complex numbers. I used to give a short history to my students about how once the decision was made to accept the square root of -1 as a number, all the rest logically followed, although it took a lot of work to figure it all out. I particular like, from an aesthetic point-of-view, the math of the nth complex roots of a number; the math of e^(ix), including Euler's identity; and the derivation of the Mandelbrot set using iterations on each point in the complex plane. However, to clarify. I didn't say prime numbers were invented: I said they were defined. Prime numbers are what they are, but someplace along the line people made a defintion (a number is prime if ...), and then started studying them. We didn't invent prime numbers themselves, but we invented (by naming and defining) the term and other symbolisms that led to us discovering the huge set of facts and properties that we are now aware of. And to kf: Wikipedia says that hash marks are a popular hypothesis about the origin of Roman numerals, but that using fingers and the palm are an alternative hypothesis. I didn't know about the palm hypothesis, but I don't think deciding which hypotheiss is correct is an important issue. I've already said, twice, that it was time for me to leave this conversation: maybe third times a charm. :-)hazel_{December 16, 2018
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F/N on 78: Another capital example is the reality of primes and their many fascinating properties and patterns, which, manifestly, we discovered rather than invented. Try this, every prime from 6 on is such that on squaring will be one more than a multiple of 24, cf: https://www.youtube.com/watch?v=ZMkIiFs35HQ KFkairosfocus_{December 16, 2018
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MG, 95: >>If we believe in the Law of Excluded Middle (which I know you do, KF) then pi is a normal number or it is not. So evidently, that Law logically implies mathematical platonism!>> Interesting observation. KF PS: It is not that I believe a list of three laws and take Aristotle as authority for centrality or even experience. I see that distinct identity is central to any particular possible world. This then directly draws out the laws, which have ontological not just cognitive import. A is itself i/l/o its core characteristics that give it its character and distinction from ROW, i.e. ~A, so W = {A|~A}. Then no x in W can be A AND ~A, LNC. Likewise any x in W will be A or else ~A, not being both or neither, i.e, the X-OR is key to LEM. We are seeing how pervasive distinct identity is and how it helps us understand so much, including numbers and extensions of numbers.kairosfocus_{December 16, 2018
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H, Pardon, but I find it hard to accept your response. Let me clip (noting that H has stated that she has been a Mathematics teacher), highlighting core relevant remarks: KF, 71: >> . . . The study of the logic of structure and quantity is culturally shaped but it is also constrained by the substance of structure and quantity. As has been repeatedly highlighted that starts with the import of distinct identity [--> note, I have repeatedly pointed out how this directly brings out nullity, unity, duality and extends through order types to the naturals thence the further sets]. The issue is not the cultural framing of the study, it is the necessary entity rooted substantial core. As has also been repeatedly highlighted>> ET, 72: >>hazel- inventing numerals is not inventing mathematics.>> H, 73: >>Did we discover Arabic numerals, ET.? Or perhaps you can explain what you mean by “mathematics” if you don’t include, at least in part, the symbol systems we use to write mathematics? . . . >> H, 74: >>[N]ot quite sure what your point in 71 is, kf. FWIW, the first paragraph in the longer post I am working on starts with this: “5. As kf points out, the idea of distinct identity is the starting point of math: we start with 1. From there, with the use of logic and proper definitions, we have historically built the mathematical edifice that exists today.” [--> Nope, we start with much more that is manifestly embedded in the world. The dichotomy of distinct identity W = {A|~A} directly expresses nullity, unity, duality and so successive order types with associated incremental cardinalities.] I solidly agree with you about the fundamental importance of distinct identity as the basis of mathematics: it all starts with the unit number, and builds from there.>> H, 75: >> PS to ET: I wrote, “inventing Arabic numerals and the common arithmetic algorithms we use”, not just numerals. So, to expand my question to you, do you consider the normal way we do long division, like you learned in school, part of mathematics?>> KF, 76: >>again, recall that you are speaking with people who have routinely used hexadecimals and binary digits (including binary coded decimals) so we obviously had to spend a fair deal of effort to learn different systems for numerals and even different processes such as use of twos or ones complements etc. We know the difference between the substance of structure and quantity — i.e. Mathematics — which is embedded in reality and our partly culturally shaped study of it; the discipline of Mathematics. Such is a very specific and pivotal distinction. This is clearly coming down to recognising (or refusing to recognise) that truth is the accurate description of reality. There is mathematical reality and mathematical truths use culturally conditioned symbol systems AKA codes (yes, language), to describe such then to do analysis, procedures, calculation etc.>> KF, 78: >>No, again, distinct identity PRESENTS us with nullity, unity and duality. Surely A is one thing and ~A is another; two-ness is patently present in any distinctly identifiable world . . . just as a beginning. This structure then presents us with succession from nullity, which we can analyse and term order type and thus we find the naturals. The transfinites then arise from the order type of the naturals and so forth. Something, which we were restrained from recognising by our cultural influences, even though the fact that naturals will endlessly exceed any given natural we state or symbolise say as k then exceed k+1, k+2 . . . etc as though k were only 0, is trivially demonstrable, showing a new domain of quantity, the transfinite. All of this is inherent in the rational framework of any distinct world. The work of von Neumann et al did not invent natural counting numbers. Simply the five fingers on our hands confronts us with cardinalities from 0 to 5, with the reality of equality, greater or lesser cardinalities and much more. Refusal to attend to manifest realities of the world does not transmute them into cultural artifacts just because we may study and name them, creating symbol systems to analyse, describe and use such realities. Another capital example is the reality of primes and their many fascinating properties and patterns, which, manifestly, we discovered rather than invented. However, such persistent refusal is symptomatic of cultural influences that are telling us about the state of a civilisation that increasingly throws objective truth overboard.>> ET, 77: >> I will grant only that we may have invented the symbols we use. But without mathematics this universe would not exist. Read comments 45 [Tegmark's universe made of Math thesis and discussion clipped above by me at 69] and 68 ["It is obvious that mathematics was discovered. Just ask the people who did the discovering or read what they say- start with Srinivasa Ramanujan. . . . "]>> KF, 81: >>ET, the progression from |, ||, |||, ||||, V (for the spread out palm) to 1,2,3,4, 5 is obvious and cultural. The invention of the abacus invites the place value notation and the open-bead for the empty place. Thence, power series representation on a base and the decimal point marker for whole and fractional parts. I suspect most who use decimal numbers don’t realise that they are using a compressed form for power series, where the convergence on the fractional side and the implication of endless continuation brings in the irrationals and thus the continuum. Which is another whole quantitative domain that reality confronts us with. Multidimensional continuum then allows us to see the structure and quantities of space and of shapes, lines and locations in space, including curves. Whole worlds of necessarily present properties appear. I already pointed out the use of ropes with twelve evenly spaced knots to specify a right angle. Why right angles and up/down and horizontal directions are important then comes up, embedded in properties of a terrestrial planet. At much more sophisticated level, we may STUDY how gravitation reflects warping of spacetime, and much more. The blindness to how deeply structure and quantity are embedded in reality is a sobering symptom of where our civilisation is headed now that it is hell bent on embracing inherently irrational worldviews.>> H, 79: >>I think it is important to try to make clear distinctions when discussing issues, and the points I made in 70 were meant to be about pure mathematics, not about the larger issue of mathematics in the physical world. [--> the original context of EG's challenge was a physical case, a stone falling off a cliff, but it should be obvious from ET's reference to Ramanujan that number theory is a primary reference, also if Mathematics is embedded in the world it will necessarily come out in its physics] So perhaps we can agree that the humans have invented the symbolic system that we use for representing mathematics, including symbols and definitions. [--> Notice the continued avoidance of the other half of the story, the extent of the embedded substance of Maths, starting with numbers beyond unity. I remind, distinct identity of world W leads to W = {A|~A}, thus nullity, unity and duality, two distinct entities being present in the partition and no-thing between or beyond. This already grounds the discovery of successive, endless order types per von Neumann.] Would you agree to this broader statement? >> [--> FAIR COMMENT: half the story, omitting a material part and studiously avoiding direct substantial interaction with me who has put it in play right from explaining why I find the dual character of the definition of Mathematics significant. Namely, Mathematics is [the study of] the logic of structure and quantity. The parenthesis marks the substance vs study dichotomy, the substance being in the world and discovered, the study engaging the substance and using culturally influenced symbols, axiomatisations, codes, procedures etc. This should not have to be belaboured but that is where we are.] H, 80: >>kf, to what does “no, again” refer in 78? Or does your post have anything to do with what I have written? [--> subtext of evasion]>> ET, 82: >>we may have invented the symbols. That is all I will grant. What did Ramanujan discover? Was it the equations fully equipped with the symbols he didn’t know? Or did he project the known symbols onto the equations he was given? For me I would say that we have discovered it all. That is because all of the information has been loaded into this universe.>> KF, 83: >>H, I spoke to your recent comments, and have made sufficiently specific remarks that that may be readily discerned. [--> as in study vs substance, distinct identity directly indicating 0, 1, 2 and thence successive order types embracing the Naturals etc. Notice also, how often I have used the extension to reals and continuum to highlight the structure-quantity aspects of space, even in an earlier thread speaking to moving fingers across space to type comments]>> H, 85: >> in 78, kf wrote, Another capital example [of what I am not sure] is the reality of primes and their many fascinating properties and patterns, which, manifestly, we discovered rather than invented. I think if you were to re-read 70, you would see that I would agree with you that, once prime number is defined, “their many fascinating properties and patterns” were discovered, not invented. We are in agreement on that issue.>> [--> further studious evasion of substance vs subject and the extent to which structure and quantity are embedded in a spatio-temporal world in which we type comments by moving fingers and rocks fall off cliffs at rates of change of displacement exhibiting rates of change of such velocity as are manifest in acceleration tied to force-inertia balances] H, 88: >> Well, I didn’t quit. I think I understand what kf is trying to say. [--> notice condescending tone, I am not merely inchoately attempting like a spermologos picking up ill-digested scraps around the Agora] I have been talking about pure mathematics, and I agreed that recognizing the important of distinct identity, the unit one, is the foundation upon which all mathematics is built. [--> Notice the repeated studious evasion of the inconvenient 0, 1, 2 and implied onward succession to N in W = {A|~A}] His point is that we experience distinct identity in the world: five fingers, or a piles of stones. This is true, and is the underlying experience which motivated the very beginning of pure mathematics. [--> Notice, again, evasion of substance vs study] The history of this is very interesting, and I have written about it in the past (although I don’t know if I’ve done that here.) Before a written system was developed, there is anthropological evidence that people used a one-to-one correspondence between a group of pebbles and some other group, such as a herd of sheep, to compare quantities. [--> thus, equality is embedded in reality, manifest in matching set cardinalities] Eventually simple hash marks substituted for the pebbles – the first written numbers were simply vertical hash marks with a slash through every four to represent a group of five. [--> Tally-marks are not to be equated to original form Roman Numerals which use the V to show the span of five fingers] If [--> the loaded if] pointing to this is what kf is trying to do [--> In this context I explicitly highlighted Roman Numerals, as they are the most blatant case in point of using cardinalities manifest in our fingers and toes] , then I agree. We started the building of pure mathematics by inventing ways to write about the simple distinct identities and quantities that we observed in the world around us. [--> again, evades discussing the import that much more than 1 is embedded in the world, i.e. our world is imbued with, pervaded by manifest, multiple forms of structure and quantity, manifesting the core substance of Mathematics]>> H, 90: >>Taking ET seriously [--> by contrast with . . . ], pure mathematics is mathematics done within a logical system, without reference to any application or reference to things outside the system. [--> Oh, what was all of that silly stuff on abstract logic-model worlds that are poossible worlds that can connect to other worlds through family resemblance or even through discovery of necessary entities framework to all worlds? Oh, nothing worth noticing . . . ] For instance, I think the vast majority, if not all, of the work done by Ramanujan, whom you referenced, was in pure mathematics. Did you scroll through the Wikipedia article and see some of them: that’s pure mathematics. Some of his discoveries may have found later application (I don’t know if that is true or not), but my informed guess that most of them have no relevance to anything outside pure mathematics. >> [--> R's work was a materially independent discovery of and in parts extension to number theory, which brings out properties of the set N which draws out from distinct identity through 0,1,2 and order type succession, thus is a transfinite set of necessary beings embedded in the framework of any possible world] ET, 91: >>Discoveries is the operative word, hazel.>> H, 92: >>Yes, I know. Re-read post 70. I think I’ve been clear about what part of pure mathematics is invented and which part discovered. >> [--> Just the opposite, H suppressed the extent and import of what is discovered. Not to mention, this evades conceding explicitly that the distinction between embedded locked in the world substance and culturally influenced study that gains objectivity by having to correspond to that substance is fully justified.] The rhetorical patterns and the balance on substantial merits now stands clearly revealed through this evasive half-concession. KFkairosfocus_{December 16, 2018
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MG, of course, while I studied Maths itself, much of my interest was on the side of systems, where the complex frequency domain in its various guises is a huge domain of praxis. Thus i (or j) becomes a key to seeing the frequency and transient structure of systems behaviour thanks to Laplace and Fourier. Spotting poles and zeroes can be a bit of a sport, whether you think s-domain or the linked z-domain. The latter of course being particularly suited to discrete-state systems. This brings out how a logic model world (ultimately underpinned by relevant axioms, postulates etc) can function as a possible albeit abstract world that then highlights things that may be necessary entities present in all worlds or just things present in worlds in the neighbourhood or family of the one we live in. But underlying, we see that once inertia, dynamics, energy storage, friction and the like or equivalent are present so differential equations or difference equations capture behaviour, just the structure of the equations reveals a lot about transient and frequency-oriented behaviour. Again, substance of structure and quantity are pivotal. I also tend to use j as an orthogonal rotation operator which then naturally brings out how sqrt-(-1) is j: j*x --> y, j*y --> -x, i.e. -1*x so j = sqrt-(-1). This then naturally brings in rotation, phasors and frequency. So, complex numbers are actually a way to do vectors algebraically, as the extension to the ijk basis further illustrates. KFkairosfocus_{December 15, 2018
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math guy @ 93,
Were the complex numbers (the field C, not the symbolic representations x+iy) non-existent until Tartaglia “invented” i, after which they sprang fully formed into existence like Athena out of Zeus’ head? Now a very interesting thing about these complex numbers is their relation to the distribution of prime numbers. The Riemann Hypothesis says that a certain regularity result about the distribution of primes is equivalent to a complicated complex-valued function of a complex variable having its roots lie solely on the line 1/2+yi. Why should Tartaglia’s “invention” dictate behavior of prime numbers?
Or, if we invented math, does that mean we invented irrational numbers like pi and the square root of 2? Apparently it was the Pythagoreans who “discovered” irrational or alogos numbers. In fact, the discovery was credited to a guy by the name of Hippasus, though according to legend they were not too happy about his discovery and while out at sea, they threw him overboard.john_a_designer_{December 15, 2018
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KF, Regarding the decimal expansion of pi, another famous conjecture is that pi is a "normal" number. This means that taking any sufficiently large initial segment of its decimal expansion, the digit "7" will occupy 1/10 of the segment, the string "48" will occupy 1/100 of segment, and so on. In particular any finite string of n integers will appear in the expansion, and at proportion 10^{-n} for sufficiently large initial segments of the expansion of pi. If we believe in the Law of Excluded Middle (which I know you do, KF) then pi is a normal number or it is not. So evidently, that Law logically implies mathematical platonism!math guy_{December 15, 2018
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WJM has hypothesized that mathematical symbols are what we humans invent. Fields Medalist (i.e. really smart guy) Allain Connes suggests that humans invent formal systems, or at least the axioms thereof, as the tools used to investigate what he calls primordial mathematics (the relationships and structures that seem inherent to platonists). A-mats claim to debunk platonism as follows (idea first espoused by Quine): suppose there is a platonic realm containing all of primordial mathematics (and perhaps lots more). It is of course immaterial and not part of the universe we occupy, since everything here is only imperfect shadows of the platonic realm which decay and die. Then how do we as purely material entities contact or perceive the platonic realm? I see primordial mathematics as a striking counterexample to materialism. Since I myself DO get occasional glimpses of the platonic realm of primordial mathematics, I am convinced it is real and therefore materialism cannot be true. My subjective experience is evidence (to me) that humans are more than machines made of meat; that we do have some non-physical contact with an eternal unchanging reality.math guy_{December 15, 2018
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Another phenomenon (already hinted at by hazel) is that "inventing" some mathematical structure (she mentions prime numbers) invariably leads to all sorts of unforseen consequences and subsidiary facts about the structure, only established by mathematical proofs. A classic example is the imaginary unit i, first postulated as the "imaginary unit" by Tartaglia, Cardano, et al. This was a crutch used to find roots of cubic and quartic polynomials. What about higher order polynomials? Do we need new "imaginary" numbers to find their roots? No, Gauss, et al proved that the complex field C is algebraicly closed, meaning that the roots of any polynomial with coefficients in C also lie in C = R(i). Were the complex numbers (the field C, not the symbolic representations x+iy) non-existent until Tartaglia "invented" i, after which they sprang fully formed into existence like Athena out of Zeus' head? Now a very interesting thing about these complex numbers is their relation to the distribution of prime numbers. The Riemann Hypothesis says that a certain regularity result about the distribution of primes is equivalent to a complicated complex-valued function of a complex variable having its roots lie solely on the line 1/2+yi. Why should Tartaglia's "invention" dictate behavior of prime numbers?math guy_{December 15, 2018
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Yes, I know. Re-read post 70. I think I've been clear about what part of pure mathematics is invented and which part discovered.hazel_{December 15, 2018
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Discoveries is the operative word, hazel.ET_{December 15, 2018
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Taking ET seriously, pure mathematics is mathematics done within a logical system, without reference to any application or reference to things outside the system. For instance, I think the vast majority, if not all, of the work done by Ramanujan, whom you referenced, was in pure mathematics. Did you scroll through the Wikipedia article and see some of them: that's pure mathematics. Some of his discoveries may have found later application (I don't know if that is true or not), but my informed guess that most of them have no relevance to anything outside pure mathematics.hazel_{December 15, 2018
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"Pure mathematics"? Is that what you get when you run mathematics through a series of filters? :cool:ET_{December 15, 2018
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Well, I didn't quit. I think I understand what kf is trying to say. I have been talking about pure mathematics, and I agreed that recognizing the important of distinct identity, the unit one, is the foundation upon which all mathematics is built. His point is that we experience distinct identity in the world: five fingers, or a piles of stones. This is true, and is the underlying experience which motivated the very beginning of pure mathematics. The history of this is very interesting, and I have written about it in the past (although I don't know if I've done that here.) Before a written system was developed, there is anthropological evidence that people used a one-to-one correspondence between a group of pebbles and some other group, such as a herd of sheep, to compare quantities. Eventually simple hash marks substituted for the pebbles - the first written numbers were simply vertical hash marks with a slash through every four to represent a group of five. If pointing to this is what kf is trying to do, then I agree. We started the building of pure mathematics by inventing ways to write about the simple distinct identities and quantities that we observed in the world around us.hazel_{December 15, 2018
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kf, I don't "readily discern" at all how what you wrote responds to the specific things I've written. But I will quit trying if you can say no more.hazel_{December 15, 2018
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As far as whether or not any particular person "finds the arguments compelling", what possible difference does that make? One either has a rational criticism based on fact, evidence and/or logic of the arguments presented, or they do not. One either has a rational argument for the idea that humans invented mathematics and did not discover it, or they do not.William J Murray_{December 15, 2018
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in 78, kf wrote,
Another capital example [of what I am not sure] is the reality of primes and their many fascinating properties and patterns, which, manifestly, we discovered rather than invented.
I think if you were to re-read 70, you would see that I would agree with you that, once prime number is defined, "their many fascinating properties and patterns" were discovered, not invented. We are in agreement on that issue.hazel_{December 15, 2018
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Ed George said:
William felt that my argument was sufficient to warrant a dedicated OP. Maybe you should take it up with him.
Incorrect. As far as I can tell, you've made no argument for your position. Perhaps that is what hazel is attempting to do with that person's latest posts, I don't know. What I felt was that your comments and criticisms were based on a failure to understand the argument being made in favor of what mathematics represents being a discoverable, inherent aspect of the universe. My post was about presenting that argument in a different light in order to try and provide understanding. So far, I haven't seen anyone answer or address this very simple question pertaining to whether we invent or discover mathematics: Did humans discover that 2+3=5? Or did they invent it? I’m not talking about the symbols; I’m talking about what the symbols refer to. For example, did we just invent the fact that 2 of anything plus 3 of the same thing equals five of that thing? William J Murray_{December 15, 2018
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H, I spoke to your recent comments, and have made sufficiently specific remarks that that may be readily discerned. KFkairosfocus_{December 15, 2018
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hazel, we may have invented the symbols. That is all I will grant. What did Ramanujan discover? Was it the equations fully equipped with the symbols he didn't know? Or did he project the known symbols onto the equations he was given? For me I would say that we have discovered it all. That is because all of the information has been loaded into this universe. Starlings, held in captivity never seeing the night sky, become restless and wanting to migrate when shown the night sky that would trigger their normal migration. "Instinct" is juts a word for "we have no idea". For me they are tapping into the Starling information vault. ("See "Why is a Fly Not a Horse?")ET_{December 15, 2018
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ET, the progression from |, ||, |||, ||||, V (for the spread out palm) to 1,2,3,4, 5 is obvious and cultural. The invention of the abacus invites the place value notation and the open-bead for the empty place. Thence, power series representation on a base and the decimal point marker for whole and fractional parts. I suspect most who use decimal numbers don't realise that they are using a compressed form for power series, where the convergence on the fractional side and the implication of endless continuation brings in the irrationals and thus the continuum. Which is another whole quantitative domain that reality confronts us with. Multidimensional continuum then allows us to see the structure and quantities of space and of shapes, lines and locations in space, including curves. Whole worlds of necessarily present properties appear. I already pointed out the use of ropes with twelve evenly spaced knots to specify a right angle. Why right angles and up/down and horizontal directions are important then comes up, embedded in properties of a terrestrial planet. At much more sophisticated level, we may STUDY how gravitation reflects warping of spacetime, and much more. The blindness to how deeply structure and quantity are embedded in reality is a sobering symptom of where our civilisation is headed now that it is hell bent on embracing inherently irrational worldviews. KFkairosfocus_{December 15, 2018
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kf, to what does "no, again" refer in 78? Or does your post have anything to do with what I have written?hazel_{December 15, 2018
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Thanks, ET. I think it is important to try to make clear distinctions when discussing issues, and the points I made in 70 were meant to be about pure mathematics, not about the larger issue of mathematics in the physical world. So perhaps we can agree that the humans have invented the symbolic system that we use for representing mathematics, including symbols and definitions. Would you agree to this broader statement?hazel_{December 15, 2018
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PS: No, again, distinct identity PRESENTS us with nullity, unity and duality. Surely A is one thing and ~A is another; two-ness is patently present in any distinctly identifiable world . . . just as a beginning. This structure then presents us with succession from nullity, which we can analyse and term order type and thus we find the naturals. The transfinites then arise from the order type of the naturals and so forth. Something, which we were restrained from recognising by our cultural influences, even though the fact that naturals will endlessly exceed any given natural we state or symbolise say as k then exceed k+1, k+2 . . . etc as though k were only 0, is trivially demonstrable, showing a new domain of quantity, the transfinite. All of this is inherent in the rational framework of any distinct world. The work of von Neumann et al did not invent natural counting numbers. Simply the five fingers on our hands confronts us with cardinalities from 0 to 5, with the reality of equality, greater or lesser cardinalities and much more. Refusal to attend to manifest realities of the world does not transmute them into cultural artifacts just because we may study and name them, creating symbol systems to analyse, describe and use such realities. Another capital example is the reality of primes and their many fascinating properties and patterns, which, manifestly, we discovered rather than invented. However, such persistent refusal is symptomatic of cultural influences that are telling us about the state of a civilisation that increasingly throws objective truth overboard.kairosfocus_{December 15, 2018
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hazel, I will grant only that we may have invented the symbols we use. But without mathematics this universe would not exist. Read comments 45 and 68ET_{December 15, 2018
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H, again, recall that you are speaking with people who have routinely used hexadecimals and binary digits (including binary coded decimals) so we obviously had to spend a fair deal of effort to learn different systems for numerals and even different processes such as use of twos or ones complements etc. We know the difference between the substance of structure and quantity -- i.e. Mathematics -- which is embedded in reality and our partly culturally shaped study of it; the discipline of Mathematics. Such is a very specific and pivotal distinction. This is clearly coming down to recognising (or refusing to recognise) that truth is the accurate description of reality. There is mathematical reality and mathematical truths use culturally conditioned symbol systems AKA codes (yes, language), to describe such then to do analysis, procedures, calculation etc. KFkairosfocus_{December 15, 2018
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PS to ET: I wrote, "inventing Arabic numerals and the common arithmetic algorithms we use", not just numerals. So, to expand my question to you, do you consider the normal way we do long division, like you learned in school, part of mathematics?hazel_{December 15, 2018
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ot quite sure what your point in 71 is, kf. FWIW, the first paragraph in the longer post I am working on starts with this: "5. As kf points out, the idea of distinct identity is the starting point of math: we start with 1. From there, with the use of logic and proper definitions, we have historically built the mathematical edifice that exists today." I solidly agree with you about the fundamental importance of distinct identity as the basis of mathematics: it all starts with the unit number, and builds from there. So, I'm curious what your thoughts are on the points I made at 70? Is there anything there you disagree with?hazel_{December 15, 2018
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Did we discover Arabic numerals, ET.? Or perhaps you can explain what you mean by "mathematics" if you don't include, at least in part, the symbol systems we use to write mathematics? Also, are there other parts of my post at 70 that you agree or disagree with? I'm curious to know some of what you think about this part of the topic.hazel_{December 15, 2018
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hazel- inventing numerals is not inventing mathematics.ET_{December 15, 2018
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