At PBS: Puzzle of mathematics is more complex than we sometimes think

_{Denyse O'Leary

April 19, 2015

Mathematics, News}

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Astrophysicist Mario Livio shares some thoughts: Math: Discovered, Invented, or Both?

The puzzle of the power of mathematics is in fact even more complex than the above examples from electromagnetism might suggest. There are actually two facets to the “unreasonable effectiveness,” one that I call active and another that I dub passive. The active facet refers to the fact that when scientists attempt to light their way through the labyrinth of natural phenomena, they use mathematics as their torch. In other words, at least some of the laws of nature are formulated in directly applicable mathematical terms. The mathematical entities, relations, and equations used in those laws were developed for a specific application. Newton, for instance, formulated the branch of mathematics known as calculus because he needed this tool for capturing motion and change, breaking them up into tiny frame-by-frame sequences. Similarly, string theorists today often develop the mathematical machinery they need.

Passive effectiveness, on the other hand, refers to cases in which mathematicians developed abstract branches of mathematics with absolutely no applications in mind; yet decades, or sometimes centuries later, physicists discovered that those theories provided necessary mathematical underpinnings for physical phenomena. Examples of passive effectiveness abound. Mathematician Bernhard Riemann, for example, discussed in the 1850s new types of geometries that you would encounter on surfaces curved like a sphere or a saddle (instead of the flat plane geometry that we learn in school). Then, when Einstein formulated his theory of General Relativity (in 1915), Riemann’s geometries turned out to be precisely the tool he needed! More.

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Comments

KF,
DS, I am sorry, I did not just list the names Fuzzy Logic and Non-Euclidean Geometry. I specifically indicated ways in which the former diverges from normal set membership, and how the latter, right from axioms is deliberately divergent from Euclidean Geometry on the parallel lines postulate or axiom.
:-? I'm also sorry, because that doesn't address my question. No doubt if you try to merge the axioms for Euclidean and hyperbolic geometry, you will get an inconsistent system, so the two systems are incompatible in that sense, and similarly for fuzzy logic. Let me reflect back to you your argument as I understand it:
We find a more than expected number of connections in mathematics, exemplified for instance by the nexus of Euler's identity. This suggests the existence of some "root mind" ordering reality and assisting us in the development of our mathematics. In the above, "more than expected" means "more than expected under the null hypothesis that there is no root mind ordering reality and that mathematics is purely invented by humans".
Is that correct? I understand that you do believe that our current mathematical systems have been developed with the help of God (including non-Euclidean geometries and fuzzy logic perhaps?). And we do see many interesting connections, as you have pointed out. My challenge is for you to demonstrate that, assuming that God had withheld His assistance, and humans were forced to invent mathematics on their own, that there would be fewer of these connections (which is obviously an impossible task, but how else could you proceed?).daveS_{April 22, 2015
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DS, I am sorry, I did not just list the names Fuzzy Logic and Non-Euclidean Geometry. I specifically indicated ways in which the former diverges from normal set membership, and how the latter, right from axioms is deliberately divergent from Euclidean Geometry on the parallel lines postulate or axiom. Where, it is implicit -- but should now be pointed out -- that set theory is foundational to modern mathematics and the discovery of non euclidean geometries as full bore axiomatic systems were a key start point for modern mathematics. The analogy of cars sharing a common general architecture but having divergent mutually inconsistent parts is thus seen to be apt. By contrast, what happened with the eqn 0 = 1 + e^i*pi, was an astonishing convergence of whole domains that then flooded out again in a vast array of areas directly relevant to so much of reality. I just note how one of the linked articles points out how the transcendental numbers e and pi, coming from utterly different contexts and with all the strange properties of being transcendental, now turn out to be in perfect lockstep not just in a few digits but to infinity . . . that alone should give us serious pause to think on this eqn and what it is saying. As to raising e to the i*pith power, that is in fact the crux as through this we access the world of phasors and rotating systems, also onwards Fourier and Laplace transforms thence differential and difference eqns, systems, signals, responses, system dynamics and hugely more. This fully justifies the point that the form e^i*pi = -1 is also shocking. Not to mention the quantum world. It is no accident that Physicists voted it as comparable to Maxwell's four eqns for electromagnetism, which have had the same sort of power of synthesis. Newtonian Dynamics and the like are comparable. The difference is, Euler's expression in context is wholly mathematical, will never be superseded, and shows an astonishing unity of both mathematics and the real world. Fourier, Laplace, Z transforms, differential and difference eqns, signals and systems etc etc etc. A vast sweep lurking as the berg beneath that simple little expression and its immediate context, a direct context accessible to a 6th former. KFkairosfocus_{April 22, 2015
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DS, Kindly note my observation, some.
Yes, thanks for clarifying.daveS_{April 22, 2015
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DS, Kindly note my observation, some. In this thread I have had to deal with outing, personal attack and indirect personal attack meant to associate me as an advisor with a policy made to sound foolish by omitting highly material and easily ascertained facts that have been in global news headlines in many cases but which may now have faded from public memory. Just scroll up. The matter is so bad -- and so bad in connexion with another attack in another forum emanating from much the same circles -- that I have had to lay out more on my background than I would have liked. KFkairosfocus_{April 22, 2015
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Yikes. "Toxic points and insinuations"? I certainly didn't have anything like that in mind while typing my posts. The last part of my post #69 describes the specific question I have. Just pointing to fuzzy logic and non-Euclidean geometry doesn't really address it. And to be clear, of course I understand the value of working with a variety of axiom systems, even those which are "nonstandard". I'm all in favor of that, and obviously it has been productive.daveS_{April 22, 2015
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DS, Pardon me but this takes on the face of a case of successive tangents that ends in pointless distraction. In reply to Jerad, I pointed to a striking case of the convergence and broad sweep of implications of the Euler eqn. I had also pointed out that had Math in general been instead like a collection of different models, you would not be likely to see the sort of deep coherence within math and onwards into experienced reality. Ever since all sorts of tangents, some loaded with toxic points and insinuations. On your request I have provided now two cases of the divergent models effect, fuzzy logic and non Euclidean Geometry which shows sharp divergence precisely on the parallel lines/angle sum triangle axiom. I have shown my point, making a reasonable contrast to two families of the sort of divergence that car parts show. This contrasts with what happened surrounding Euler's 1748 breakthrough. Now, I see a manufactured, projected "duty" to provide axioms, connections and so forth, doubtless on the assumption that if I fail to produce what would in effect be a collection of monographs, my main point can be dismissed. Not so. I have done what is needed, and have illustrated that a core part of relevant Math reflects a deep unity in the world and in math, which aptly correspond. By contrast, other things do not have that character. My point is made and in that core area there is good reason to infer that we are discovering a core deep coherence in reality that a very good candidate explanation is that said reality comes from a deeply logical-mathematical designing mind. KF PS: I do not include general logic as an example as it is better seen as pre-math, philosophy. And yes, that also has diverse possible constructed frameworks.kairosfocus_{April 22, 2015
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KF,
DS, I believe I have provided enough for you to follow up if you wish to explore the issue of limited mathematical modelling that does not typically have the sort of deep, powerful, broad, vista-opening coherence and power we see from the Euler expression in context. We do live in an age of Google search.
And I do appreciate the discussion. In fact I had done some googling, but didn't find anything relating to the number of connections in a fuzzy logic-based system. I think it's your responsibility to provide that information. I also think that's an impossible task---you simply can't make the kind of comparison necessary to support your claim about the relative impoverishment of "invented" mathematics.
The point I am making, is that certain facets of Mathematics show the sort of coherence, beauty and power that this case we are focussed on as a chief exemplar of Wigner’s remark on the astonishing effectiveness of Mathematics in the physical sciences, exemplifies. It matters not to that point that other areas show more of the modelling approach and divergences that I have compared to how car models have similar general architecture but utterly divergent parts. I think we need to redevelop a sense of wonder and appreciation that acknowledges the sheer, raw beauty and power of such things. You give me reason to appreciate my conversations with former students who spoke of how, long years after studying Control Systems together, they were still finding themselves pole spotting from time domain damped oscillatory behaviour, on the heavy rubber sheet model. Which, BTW is very much connected back to Euler’s work. KF
I share the same sense of wonder. I like mathematics as much as the next person. It's the coolest thing ever, IMO. In fact, I probably spend more time than I should reading about it. But in the end, I agree with Don Pedro in #77.daveS_{April 22, 2015
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DS, Yes, especially projective, Spherical and of course Riemannian; where the divergences on the parallel lines postulate illustrate the different car parts analogy rather aptly. None of that detracts from the force of the shocking convergence and breakthrough connexions that lurk in the Euler expression. KFkairosfocus_{April 22, 2015
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Querius,
So in relation to your question, what did you conclude from your non-Euclidean geometry class(es)?
Well, I didn't really attempt to quantify the number of connections corresponding to any particular choice of axioms. However, it's clear that you can get interesting and useful mathematics even in a non-Euclidean setting.daveS_{April 22, 2015
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Q, I think you have a point on paradigms, shifts and how there is a gap across the paradigms that leads to living in different mind spaces. The cramping nature of a priori evolutionary materialism is on shocking display. KF PS: I cannot but notice how AR walked off without any response once I gave the bigger picture on challenges facing a volcano disaster ravaged 350+ years standing overseas territory of the UK. GBP 300 mn plus and a Premier that -- like Oliver Twist -- asked for more, makes much more sense when we get a bigger picture, nuh. There's that old Jewish Mom's saying about half the truth said in misleading disregard to the real, whole material truth. (I think Daily Telegraph, Daily Mail etc have not been giving a true and fair view. And, I would suggest that per the Millennium Goals folks on how 0.7% of GDP on well done dev't aid would make a huge cumulative difference to global poverty and development challenges, that such aid costs a whole lot less than the blood, treasure and devastation of war. Yes, waste, fraud and corruption are challenges that need to be faced and dealt with. That's why good governance is a big issue, and one that just happens to be on my plate along with sustainability of development just mow. This BTW is part of how I found it such a breakthrough to move to AS-AD and then tie in Garrison's integration of the Hayek multi-year, multi-phase investment triangle with a C vs I production possibilities frontier simplified view of macro and business cycles that then aligns with a loanable funds model of the price to rent investment money, i.e. rate of "interest." Bring in the Schumpeter-Kondratiev 30 - 70 y long wave tech-driven cycle going back to the Sung dynasty in China in the 900's with invention of printing there; interpret 2007 on as a LW generational trough. Thence, go where Austrians tend not to like going: tickling the dragon's tail on Industrial Policy and long term capacity building to spark economic transformation. See what I mean by transforming insight? Paradigm shift is about right, had not thought about it that way before but you are right Q. It may help for me to say, by way of getting some air cleared of toxic smoke of burning ad hominem laced strawmen, I helped draft the UN's book on Capacity Development for SIDS, some years back. As coming at the problem with experience of strategic curriculum architecting and facilitation for national development pivotal Engineering degree programmes. On which, I championed fusion of Mechatronics and Info and Communication Technologies. That by itself would tell you a lot on why I think the Euler expression and linked themes are so pivotal. And, notice, how all of this fits into a common, strategic concept space that can be mapped with coherent connexions. My billet is integrated, transformational strategic change driven by pivotal insights. And yes, that is several paradigm shifts away from conventional thinking cramped by evolutionary materialist scientism. Time to think out of the box, folks!) PPS: Here's some of my thought on economics and policy: https://docs.google.com/file/d/0ByZKFBHV9ve0cUpoUW9sb2pONnc/edit On industrial renewal: http://www.angelfire.com/pro/kairosfocus/resources/mechatronics_ja.htm Here is thought on education: https://docs.google.com/file/d/0ByZKFBHV9ve0RWktQ2t4bVAzT1U/editkairosfocus_{April 22, 2015
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DP, Interesting points. I think the key issue is connexions. Which are pivotal to seeing the microcosm revealed. Just out of curiosity, how do you respond to the Mind/Concept Map idea: http://en.wikipedia.org/wiki/Mind_map . (I think this is a useful test on the sort of nexus thinking issue that underlies this thread. As a hint, I love it as an ideas and connexions visualising tool that is an aid to creativity and to thinking through. In my day job so to speak, I am in the business of multi-level, multi-directional strategic, three moves plus ahead "chess." That is your move now needs to be shaped by perceived opportunities for various counter-moves and options for your onward move, etc; all, in real time and with heavy stakes on the board.) KF PS: BTW, I use concept frameworks and connexions to then guide exploration. As a curriculum designer, my favourite course architecture is a spiral web with key concept and activity anchor points and lines, that uses a spiral of learning activities to introduce and deepen exposure to a subject loop by loop that deepens insight on the key points as development proceeds. While, making use of key, nexus case studies. And, my fav development strategy is the evolutionary spiral http://en.wikipedia.org/wiki/Spiral_model and note US DoD here http://www.dau.mil/pubscats/PubsCats/PM/articles03/fark-ja03.pdfkairosfocus_{April 22, 2015
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F/N: i will say this, this thread has led me to do a couple of days searching on the issue of the one and the many, which I have always seen as central: unity amidst diversity in an evidently coherent, common world. I am finding an amazing lack of broad and deep reflection on it in current times, reflected in how deep I have had to search to find things of interest and substance. And yet it is one of the earliest concerns of Western thought, and is of deep, enduring significance today. I am thinking that this, too points where ever so many are desperate not to go, and it has become a road mainly not travelled save by the odd theologian-philosopher and one or two others. At least, that is the strong impression I am picking up. It certainly does not come up for the sort of rich and deep reflection I would expect of something that is such a nexus problem. But then, this is the age of the narrow specialist and this problem is just the opposite of the leanings of our day. Methinks De Bono on the power of broad, lateral thinking may have somewhat to say to us. I think I need to book a long, leisurely lunch date with good old Sophie, our elder sister and counsellor. And take along my newly modded 20+ year old Sheaffer fountain pen that now has about a 0.8 mm italic/calligraphy style broad nib by dint of grinding per recipe of pen enthusiasts. It did my heart good to get my fingers dirty with ink and swarf, etc. with an end product that gives me what a classic Parker 51 stub italic nib would do. Joins my new Parker 45 ball pen (I picked that up from Amazon on seeing it is no longer being made . . . lost one a decade back under suspicious circumstances) and longstanding 0.5 mechanical pencil. KFkairosfocus_{April 22, 2015
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So in relation to your question, what did you conclude from your non-Euclidean geometry class(es)?
Non-Euclidean geometry does not represent a different kind of mathematics. It is a generalisation of the notion of "geometry" (working according to the same logic and with some of the same or similar axioms). By the same token, complex numers are a generalisation of the notion of "number". Nothing new here. Fractions, and negative and irrational numbers (not to mention the concept of "zero") also had the status of "invented" extensions of classical arithmetic until they became familiar enough to be considered "normal". Complex numbers have been around for centuries now, and many people no longer regard them as esoteric. e^({pi}*i)+1=0 is an impressively elegant expression, but it doesn't mean much by itself and you can't deduce much from it. It's just a special instance of Euler's general formula, with x={pi}. The latter formula is vastly more useful and interesting, but hardly mystical. Mathematics is better served if you try to understand it thoroughly and not just gaze at selected equations in mute awe (or worse still, if you fall into flight of ideas and don't stay mute).Don Pedro_{April 22, 2015
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Q, thanks again. I see you have been sitting with Sophie and listening to her. She has much to teach us, if we will but listen. KFkairosfocus_{April 22, 2015
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DS, I believe I have provided enough for you to follow up if you wish to explore the issue of limited mathematical modelling that does not typically have the sort of deep, powerful, broad, vista-opening coherence and power we see from the Euler expression in context. We do live in an age of Google search. The point I am making, is that certain facets of Mathematics show the sort of coherence, beauty and power that this case we are focussed on as a chief exemplar of Wigner's remark on the astonishing effectiveness of Mathematics in the physical sciences, exemplifies. It matters not to that point that other areas show more of the modelling approach and divergences that I have compared to how car models have similar general architecture but utterly divergent parts. I think we need to redevelop a sense of wonder and appreciation that acknowledges the sheer, raw beauty and power of such things. You give me reason to appreciate my conversations with former students who spoke of how, long years after studying Control Systems together, they were still finding themselves pole spotting from time domain damped oscillatory behaviour, on the heavy rubber sheet model. Which, BTW is very much connected back to Euler's work. KFkairosfocus_{April 22, 2015
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Steno, First, you doubtless know why I use my consultancy persona in online discussions -- it reduces spam issues and ID theft opportunities. Your disrespectful and rude insistence on violating that speaks volumes, volumes that fit in with your general attitude as has been on display. An attitude that is further manifest from your playing the you no speaka da Inglish rite tactic, despite the fact that in 18, there was an expansion of what was said in 11 . . . with an expanded summary using an apt wiki clip, subsequently, links to two appreciations out there. My response at this point is that I do not believe you in your rhetorical stance, for more than enough has been given for one who is genuinely interested to understand. From me and from others. Q's comment speaks volumes, which you should heed. (Q, thanks.) I think the truth is, this case is one that speaks powerfully to how some major features of Mathematics open up vast vistas, showing an inner logical architecture of reality that invites reflection on where you evidently are desperate never to go: there is a divine mind's shadow on the doorstep. In Mathematics, of all places. But then, Mathematics is by nature an abstract, defiantly non-science discipline on the power of pure logic studying quantity, structure and relationships that somehow seem to govern so much of reality. A discipline that turns out to be a major necessity in Science. With it's daughter-discipline, computing, alongside. In short, your response speaks to this being one of the many chinks in the armour of today's dominant (and too often domineering) evolutionary materialist scientism and its fellow travellers. But, such from the outset, cannot coherently account for mind and its powers of knowledge, insight, reason and purpose, so that is not news. I think that the very same Euler, a Christian, has somewhat to say. I think something was clipped above, but a fuller cite is appropriate, from a letter to a Princess on the so-called free thinkers of his time:
The apostles and a multitude of Christians [--> 500+ eye witnesses, recorded about 25 years after the event while most of these were still alive with an implied invitation to check them, cf here on] unanimously agree not only that Jesus Christ rose from the dead, but also that they have seen him with their own eyes since the resurrection and that they even communicated with Him. If one has paid attention to the doctrine and to the constancy with which it been maintained [--> i.e. in the teeth of not only objections but dungeon, fire and sword or worse], one cannot say with any semblance of truth that one has believed nothing of what has been said in this regard and that it is thus an obvious lie. One would be even less likely to say that the apostles were seduced by false imagination and that their facts were nothing but an illusion. Either that or we will be forced to state that God had miraculously blinded them all at the same time in order to propagate a false doctrine . . . . The resurrection of Jesus Christ is . . . an incontestable fact, and since such a miracle can only be the work of God alone, it is thus impossible to doubt the divinity of the Savior’s mission. Consequently, the doctrine of Christ and his apostles is divine, and since its goal is our true happiness, we can be most assured of our belief in all the promises that the Gospel has made to us, both for this life and the one to come, and we can regard the Christian religion as a work of God who is tied to our salvation. It is not necessary to expand any further on these reflections, since it is impossible for anyone, once they are convinced of the resurrection of Jesus Christ, to retain the slightest doubt about the divinity of the Holy Scripture. The freethinkers cannot put forward anything plausible against this bedrock on which the divinity of the Holy Scripture firmly rests. When they are forced to turn their attentions to this, they do all they can not to address the root of the question. They resort to all manner of loopholes to change the subject and attack other items, where they claim to find incomprehensible things and even contradictions . . . . The freethinkers have yet to produce any objections that have not long been refuted most thoroughly. But since they are not motivated by the love of truth, and since they have an entirely different point of view, we should not be surprised that the best refutations count for nothing and that the weakest and most ridiculous reasoning, which has so often been shown to be baseless, is continuously repeated. If these people maintained the slightest rigor, the slightest taste for the truth, it would be quite easy to steer them away from their errors; but their tendency towards stubbornness makes this completely impossible. [From: A DEFENSE OF THE REVELATION AGAINST THE OBJECTIONS OF FREETHINKERS, BY MR. EULER FOLLOWED BY THOUGHTS BY THE AUTHOR ON RELIGION, OMITTED FROM THE LAST EDITION OF HIS LETTERS TO A PRINCESS OF GERMANY, 1805, point XXXIV on. (It is well worth the pause to read the printer's remarks on how these things came to be excluded from the generally accessible editions of the work.)]
KFkairosfocus_{April 22, 2015
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kairosfocus, First off, my first reaction to Mario Livio's observation was that he's describing the divide between theoretical (aka pure) mathematics and applied mathematics. As many mathematicians in the past have done, you expressed a sense of appreciation and astonishment at the assemblage and relationships in Euler's equation---that compact "mathematical jewel" with representative facets of addition, multiplication, exponentiation, e, pi, i, integers, irrational numbers, imaginary numbers, and a playful hint at deeper relationships within. What's not to be amazed at? But I noticed the different reactions. On one side, people had a sense of delight, astonishment, and curiosity. On the other, there was dismissiveness, suppression, and reduction. To me the second group is what I'd expect from academic arrogants, bureaucrats of books and bells who crush their students' curiosity, creativity, and enthusiasm under the weight of their need to dominate, and who leave a trail of injured survivors in their wakes. What a pity. You give people like this some poetry, and they'll alphabetize the words, count the letters, and announce that they've seen nothing new. In contrast, I once had the privilege of attending some lectures on physical chemistry many years ago. The presenter was so enthusiastic about his subject, it was infectious! His explanations and illustrations were simple, clear, and delightful. What a difference! -QQuerius_{April 22, 2015
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DaveS wrote,
I’m just trying to explore how we know that our current mathematical system(s) are much richer in connections than “invented” mathematics.
So in relation to your question, what did you conclude from your non-Euclidean geometry class(es)? -QQuerius_{April 21, 2015
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Steno, What's pretty obvious to me is your unwillingness to comprehend the simple description from kairosfocus about anticipating a paradigm shift in science. He suggests the potential for Euler's equation, considering it's fundamental nature, to be applied to the fundamental nature of the universe. No big stretch for most of us. He then used a paradigm shift in his perception of economics as an analogy. If you were really interested in understanding what kairosfocus was thinking, you'd have come up with something a better than a question that would be considered rude in any college classroom, especially the bit about your time being too valuable. -QQuerius_{April 21, 2015
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Mr. Mullings: "F/N: I note for record that this morning I invited Steno to respond to my remarks at 37 ff:" I am still awaiting a translation of your comment at 11. My time is too valuable to try to make sense of any of your sermons.stenosemella_{April 21, 2015
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KF,
Q, let’s hear from you. It will help return things to an even keel. I confess my astonishment at the refusal to acknowledge the rather obvious nexus status of the Euler expression, duly appreciated in context.
If you're referring to me, I obviously accept that Euler's formula has connections to a wide range of mathematical fields. I don't even object to the notion that some mathematical concepts are "discovered", in that I wouldn't be surprised if any sufficiently advanced community of life forms eventually forms concepts like our "integer", "rational number", and so forth. The specific part of your argument that I question is whether and how we know that our mathematical system is especially rich in connections, and how this is evidence for God.daveS_{April 21, 2015
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KF,
DS, I already did. Partial set memberships are a part of a different approach, which in many cases will contrast with distinct identity. it is possible to hold partial membership in Hot, Warm and Cold sets (to fractions that need not add up to 100%) per a Fuzzy set model sometimes used in control systems; e.g. in Air Conditioners. Membership here takes a quite different meaning from distinct identity and there can obviously be very different results . . . try plotting a Venn diagram and assigning location under such circumstances; the issue of divergence will be dramatically shown.
Divergence in what sense? To be clear, what I'm interested in seeing is a complete axiom system for an alternative mathematics, which is as good as our current system is at modeling specifically. I would expect it to support simple tasks such as counting as well as Euclidean geometry and classical mechanics at a minimum (Otherwise, I would expect the mathematicians to keep inventing until they make all this work). It should also be demonstrably less rich in connections than our current system(s). Obviously that's a tall order, and I can't expect a complete answer here. But if this can't be at least sketched out, I think it presents a problem for your argument.
In their time, irrationals, negatives and imaginary numbers were most distinctly not obvious nor a natural progression. That they turn out to come together as in the Euler expression then open up vistas, was a great surprise. Still is. KF
Sure, but I'm saying that as soon as you commit yourself to the integers, you are going to arrive at the complex numbers eventually. The fundamental notion of distance then leads to the metric topology on C and hence convergent series.daveS_{April 21, 2015
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F/N: I note for record that this morning I invited Steno to respond to my remarks at 37 ff: https://uncommondescent.com/mathematics/at-pbs-puzzle-of-mathematics-is-more-complex-than-we-sometimes-think/#comment-560307 The lack of response is duly noted. KFkairosfocus_{April 21, 2015
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Q, let's hear from you. It will help return things to an even keel. I confess my astonishment at the refusal to acknowledge the rather obvious nexus status of the Euler expression, duly appreciated in context. Which is close to the heart of the themes Wigner sounded as pointed to in the OP. For just one example I remember my amazement at seeing how the Laplace transform turned differential eqns into a visualisable framework, and then I felt like a kid in a sweetie shop when I learned about the heavy rubber sheet representation with poles and zeros and how frequency responses could be read off from that. Later, my students loved it, and years later were telling me how they were still pole spotting from t-domain behaviours esp damped oscillations. All, tying right back to what lies in the context of that one eqn. Microcosm and gateway, even wormhole. KFkairosfocus_{April 21, 2015
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DS, I already did. Partial set memberships are a part of a different approach, which in many cases will contrast with distinct identity. it is possible to hold partial membership in Hot, Warm and Cold sets (to fractions that need not add up to 100%) per a Fuzzy set model sometimes used in control systems; e.g. in Air Conditioners. Membership here takes a quite different meaning from distinct identity and there can obviously be very different results . . . try plotting a Venn diagram and assigning location under such circumstances; the issue of divergence will be dramatically shown. In their time, irrationals, negatives and imaginary numbers were most distinctly not obvious nor a natural progression. That they turn out to come together as in the Euler expression then open up vistas, was a great surprise. Still is. KFkairosfocus_{April 21, 2015
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Hi Querius,
You’ve never taken a class in non-Euclidean geometries, right? -Q
I have. But you may have misinterpreted my position. I'm just trying to explore how we know that our current mathematical system(s) are much richer in connections than "invented" mathematics.daveS_{April 21, 2015
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KF, Could you give specific examples of these axiom systems based on modeling? Here's what I'm wondering: If we have the same sorts of modeling in mind, these axioms should also lead to the natural numbers and integers. Once you have the integers, it's natural to construct its quotient field, and there's only one way to do that, resulting in the rational numbers. Of course then one considers the completion of Q (thought of as a metric space); again, this completion is unique, and we get R. If we skip this step, then we have problems. For example, in Q^2, the line y = x does not intersect the unit circle centered at the origin. Finally, the algebraic closure of R is essentially unique, so we arrive at C. You can probably see my point. How can we tinker with our current system, in an attempt at simulating "invented" mathematics, and still arrive at a system which is useful for modeling and where Euler's formula is destroyed?daveS_{April 21, 2015
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DaveS,
Does this mean that if math was an invented thing, then the axioms and/or rules of logic we would be using would be different than those we currently have? If so, how would they differ?
You've never taken a class in non-Euclidean geometries, right? -QQuerius_{April 21, 2015
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kairosfocus, Very nicely described, but apparently some of your audience is preoccupied with maintaining the darkness and cursing the light. But I enjoyed reading the pearls your posted and hope to cast some myself. ;-) -QQuerius_{April 21, 2015
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DS, again, logic is discovered, at least so far as the core principles of reason tied to distinct identity and sufficient reason are concerned -- ponder a bright red ball on a table and the world partition it imposes. There are odd sets of logic axioms out there that arguably boil down to modelling and they do lead off in odd and potentially incoherent directions; e.g. try blending partial set memberships and distinct identity some time. The key point is, that apart from a very deep unity, it would be passing strange for things like the imaginary arbitrary number sqrt(-1) to lock in with exponentials and logs, esp. the value of x such that the area under y = 1/x between 1 and x is 1, and the ratio of diameter to circumference of a circle to interlock in the astonishing way we see 0 = 1 + e^i*pi. Such does call out for explanation, as does the onward extension into Fourier, Laplace and Z transform analysis, system behaviour (differential and difference eqns . . . transfer functions, block diagram algebra and more ) and ever so much more. It is the candidate to beat, that we are seeing coherent order that we are discovering because it was built in from the outset. KFkairosfocus_{April 21, 2015
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