A new crank! And a call for examples

While I somehow never get emails from cranks, I tend to find them at the slightest provocation on the internet.  Here’s a new one: Miles Mathis, who I’ve not run into before.  He’s also a physics crank (multidisciplinary research!) so for the more mathematically minded, just jump down to Section 2, where he has some deep-seated problems with calculus (implicit differentiation seems to have caused him to label the Calculus as corrupt), and there are some other gems mixed throughout.  For instance, the page I found, on non-Euclidean geometry.

On a more productive note, I was searching the internet for things I can bring up in the class I’m teaching to point out the flaws in Euclid’s framework, that were due to him relying too much on certain bits of intuition (between, inside, continuity, etc) without spelling them out.  If you’ve got a good example, please link to it in a comment.

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About Charles Siegel

Charles Siegel is currently a postdoc at Kavli IPMU in Japan. He works on the geometry of the moduli space of curves.
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42 Responses to A new crank! And a call for examples

  1. Jay Daigle says:

    Hey, long time no see! Anyway, you might want to check the paper linked here. I found it a fascinating paper in general–about what makes math epistemologically different from sciences–but he spends some time specifically dealing with the reasons Euclid’s proofs don’t work the way modern mathematics does. Also has a huge list of references.

  2. Perhaps you have seen the old chestnut that all triangles are isosceles? Here is a nice write-up:

    http://www.mathpages.com/home/kmath392.htm

  3. Yeah, I’ve seen that one, and am thinking of showing it to the students, but am looking for more of the like, so I have choices.

  4. Pseudonym says:

    The places where Euclid’s proofs break down are degenerate cases. It’s well-known to people who work in automated theorem proving that almost all of Euclid’s proofs come with unstated nondegeneracy conditions (sometimes called “ndg conditions” because they come up so often).

    If you have access to a well-stocked university library (one that includes the Lecture Notes in Computer Science, or LNCS for short), the best reference on this is:

    S.C. Chou and X.S. Gao, “Proving statements of constructive type”, CADE11, LNCS 607 (Springer, 1992), pp 20-34.

    There’s a copy on Gao’s web site, but some of the diagrams are missing; they shouldn’t be too hard to fill in yourself.

    The paper may be hard going if your formal logic is a bit rusty, but it’s well worth reading.

  5. Anonymous says:

    My favorite Miles Mathis theory is that pi is 4.

  6. Steven says:

    Hi Charles,

    Too bad you did’nt show any arguments before dismissing Mathis as a crank. At least he takes the effort to explain the mistakes he sees in accepted theories starting from line one.

    But I must agree his theories are bad educational material for contemporary math classes. They might show the emperor has no clothes in certain cases…

    About his theory that pi equals 4: he argues that holds for certain physical cases, like an orbital equation. For geometry with current postulates there is no change.

  7. Steven, care to explain how someone who doesn’t think that the derivative of $\ln(x)$ is $\frac{1}{x}$ ISN’T a crank? In his article http://milesmathis.com/ln.html his article is entirely gibberish, he declares perfectly acceptable things (though which require a bit more justification than in calculus books, perhaps he should read an analysis book?) to be illegal, and then says some nonsense about the derivative of $\ln(x)$ by looking at it’s differences with interval 1.

    • Steven says:

      Hi Charles,

      Your choice of language and lack of arguments is revealing. Miles goes to great lenghts on his website to explain why he uses a constant differential instead of the diminishing differential. Do you have any arguments against that or do you want me to explain it to you?

      • Right near the beginning, he claims that taking a limit inside of \ln is “gloriously illegal.” This is false, we can prove that the limits \lim \ln f(x)=\ln \lim f(x) using the fact that \ln is continuous. This is an easy exercise given to students in a first course.

        His argument after that is some bizarre thing where he points out that we don’t say that two distinct limits are identical…of course we don’t, they’re different! But the example he gives bears no resemblance to what he’s talking about, because in the case above, we’re using continuity to compute a limit, in the latter, he’s just throwing \ln around for no reason.

        He then asks “why not shift it anywhere” and writes down a formula where not all of the instances of u are bound by the limit operator. I could continue, but by this point he’s demonstrated that he’s WOEFULLY ignorant of basic, undergraduate analysis and that this is the basis of his complaints. What he does after this isn’t relevant, but it continues to be as mistaken and erroneous, and with such glaring, basic errors at the start, I’m not going to go into detail with every single error he makes later.

  8. Steven says:

    Hi Charles,

    You say:

    “Right near the beginning, he claims that taking a limit inside of \ln is “gloriously illegal.” This is false, we can prove that the limits \lim \ln f(x)=\ln \lim f(x) using the fact that \ln is continuous. This is an easy exercise given to students in a first course.”

    as if that is the core of Mathis argument but it is’nt.

    He shows that the chain rule does not apply for exponential functions and that the derivative of for those functions is a misnomer. It is explained in his paper on his calculus applied to exponential functions:

    http://milesmathis.com/expon.html

    • “I don’t need to prove it for all numbers. All numbers are defined by integers, so any extension of the basic equation is true by definition.”

      This is just false, just because numbers are built from integers doesn’t mean that all problems reduce to integers. That’s like saying x^2=2 has no integer solution, so there are no solutions, because all numbers are built out of integers.

      He seems to not believe in limits, at all, as he’s decreeing that we’re not allowed to take h\to 0, which just doesn’t make sense in any mathematical way. There is a definition for limits, and this definition can be used to prove all the basic theorems about limits, including theorems about derivatives. He takes issue with the fact that the tangent line is the closest linear approximation to a function, and starts harping on “approximation” as though it were at issue in the definition of derivative. However, he somehow has come to the conclusion that using finite differences instead of derivatives gives him true rates of change, but somehow thinks that a function can’t have a rate of change at a point, because it’s a point. It’s a complete denial of the notion of velocity at any given moment of time!

      Overall, his claims are outrageous and false, phrased provocatively on purpose, and generally contradict hundreds of years of successful physics and mathematics, to give the incorrect answers to problems which have real-world consequences.

      • I just want to say, Charles, that it’s so adorable how you play into the conceit that “Steven” isn’t a sock puppet for Mathis.

      • John, I can’t PROVE that he is, but it’s certainly crossed my mind.

      • Steven says:

        Hi Charles,

        Too bad you restrict yourself to out-of-context derogatory blurbs. Miles explains very well why he uses the constant differential. Since 1 is the smallest unit on the number line there is no need to use limits if that is used as the smallest differential. If we cannot agree on that it makes no sense to discuss calculus since that will be a comedy of confusion.

        @John: I actually take your remark as a compliment :) I have no affiliation with Miles but everybody who considers him a crank I consider to be a mainstream parrot.

      • Steven, then I guess we can’t discuss calculus, because of one of two things:

        1. If what you mean by “unit” is the same as what I mean, then of COURSE one is the smallest unit, it is THE unit, and that’s entirely irrelevant.

        2. If it isn’t, then you need to define the term, and explain why \frac{1}{n} don’t count, and why you can ignore them.

        He seems to think that the study of difference equations is adequate to do physics, with no reference to differential equations, and this is false, unless you can show me some evidence that the universe is discrete. Even given that, his approach is flawed in other ways. And that’s why he’s taken seriously in places that talk about the electric universe and plasma cosmology. And before you tell me that it must be good because Alfvén was a Nobel laureate…well, he was a plasma physicist. When all you have is a hammer, everything looks like a nail, and getting a Nobel hasn’t stopped Crick from talking about eugenics and panspermia, nor Watson from making racist remarks.

  9. I like using theorem 1 from Euclid about the construction of equilateral triangles. Euclid’s proof and a good commentary can be found at http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI1.html

  10. Andi says:

    I LOVE the statement:

    “All other General Relativity problems can be solved or estimated with this method, but I will let you discover that on your own.”

    written somewhere on the page you linked to.

    • Perhaps he’s borrowing from Descartes? In his “Geometry” he said things like “I shall not stop to explain this in more detail, because I should deprive you of the pleasure of mastering it yourself.” (Stillwell’s “Mathematics and its History”)

      On the other hand, I’m pretty sure Descartes got correct answers to things…

  11. Steven says:

    Hi Charles,

    I’m just not sure what to make of your remarks. First you seem to agree that 1 is the smallest number but then you call that “entirely irrelevant”. I assume you don’t want the discussion.

    As for the difference between difference and differential, that is the same discussion. If our universe was’nt discrete it would not exist as without space between quanta no motion would be possible.

    Also you try to equate Miles with EU for lack of arguments but he explains clearly on his website where the Electric Universe proponents go wrong, similar to what he does for mainstream physics. I guess it is because you have’nt read that much at his site.

    Your “hammer and nail” remark is just as valid for mainstream physics since they fail to find the charge field in astrophysics and the gravity field in QM.

    • You’re not responding at all to what I’m saying, if you’re reading it. At no point did I say that 1 is the smallest number. It is the smallest positive integer, yes, but \frac{1}{2} is a smaller number. By “1 is THE unit” I am saying that it is the number that we use to choose a scale for all the others, so that we know what \frac{1}{2} is by its relation to 1.

      You say “If our universe was’nt [sic] discrete it would not exist as without space between quanta no motion would be possible.” This sentence is internally inconsistent. For there to be space “between” quanta (which is already a fuzzy issue, as we can’t know position arbitrarily well by Heisenberg) then we would need a continuous universe, as in a discrete universe, we can have things occupying neighboring points. However, this still doesn’t render motion impossible, and your argument seems to be some sort of garbled version of Zeno’s (long solved) paradox.

      I admit, I’ve not read his entire site, as there’s a lot of material there and it’s fairly clearly wrong, so I wasn’t aware that EU proponents are using his stuff despite him saying they’re wrong. I’ve focused on his mathematical errors as I’m more qualified there, though upon looking over some of his physics articles, I see that my lack of qualifications as a physicist in no way prevent me from finding serious flaws in almost every single one of his arguments.

      • Steven says:

        Hi Charles,

        You are picking words to make “1/2 is a smaller number” sound like a triviality, but it is’nt. First, the smallest number on the number line is 1. If you want to redefine the number line to use “1/2 as a smaller number” you forget that to define 1/2 we first need to define the relation of the number 2 to the number 1, to define the operator “/” and then redefine the number line to have unit 1/2. Since this is just a redefinition Miles chooses 1 as the smallest Δx without losing any generality.

        This also allows me to highlight the core problem of the calculus derivation. There smaller Δx’s are selected as if one is “zooming in” on a graph to determine the slope of it. Now assume we are analyzing the graph of y = x^2. If you start with a Δx of 1 and then go to a Δx of 1/2, the size of Δx scales with a 1/2 but the size of Δy scales with 1/4, so the curvature of the curve that is being analyzed has changed! This is no issue for scaling the tangent line but it leads to incorrect results for non-linear functions.

        About the continuity of the universe: we can only assume that it is continuous but we can only observe discrete differences in it at the size of the smallest quantum (the photon). The uncertainty of position has nothing to do with Heisenberg or QM as it already holds for math based on numbers.

        I think your dismissal of Miles’ work as “fairly clearly wrong” is just your gut opinion triggered by his polemic style. I think his style is quite justified by the treatment that people that don’t parrot the mainstream get, with you providing an excellent example.

      • Steven,

        I am picking words to make things that are trivial appear to be trivial. It is completely trivial that there are numbers that are positive and less than one. The method you describe can’t be used on a sequence of numbers converging to zero, only for one bounded away, and so is totally inapplicable to differentiation.

        And no, the “zooming in” is just something that we tell undergraduates to give them intuition. It’s not what’s going on, and the analogy you give is fallacious.

        You’re making a mistake claiming that there is a “smallest” quantum, without defining what “smallest” means…according to physicists, they all behave as point masses, and yes, photons have zero mass, but so do some others. And the uncertainty principle holds because matrices that don’t commute can’t be simultaneously diagonalized. It has everything to do with noncommutative algebra, and nothing whatsoever to do with the standard numbers, which do commute.

        My dismissal of Mathis’ work is due to the fact that he makes mistakes that even most dumb undergraduates don’t make, and simultaneously insults those who have devoted their lives to the study of these objects. Now, if he has something interesting to contribute about finite difference equation, he should write it up in proper language and publish it. If he can REALLY justify his work to overthrow the current paradigm in physics, he should break it into digestible chunks that people won’t just dismiss as insane, and publish that. And if he wants to do that, he should make concrete physical predictions that are consistent with the known evidence and predict new phenomena in some manner (keeping in mind that current models are known to be accurate to quite a few decimal places).

        The fact that he doesn’t do these things indicates that either he doesn’t REALLY believe he’s right, or else that he’s actually wrong.

        In conclusion, of this conversation, he’s making extraordinary claims on the poorest of evidence, and I’m done wasting my brain power discussing his incredibly incorrect work here. Unless something drastically new and actually interesting/impressive/correct rigorous proof is brought to my attention, this is the last comment I’ll be making on this.

      • Benny says:

        If the smallest unit is taken as 1, won’t the derivative of a non-linear function change if we decide to use a different set of units? Say, we compute the velocity in metre per second instead of metre per millisecond. How is this discrepancy handled?

  12. Steven says:

    Hi Charles,

    If you want to leave the last word on my side, that’s fine with me. Apparently you seem to think that discussing basics is below the dignity of a graduate student but it is exactly Miles’ point that a lot of advanced math is just plastering over holes in the basic assumptions.

    To define numbers smaller than one, we first need to define numbers larger than one. And your remarks about sequencing also do not apply. Compared to 1, the number 1/2 converges as fast to zero as the number 2 converges to infinity, which is never.

    Since we cannot even agree on numbering basics even before we get to calculus maybe we should start at Mathis’ comments on irrational/real numbers vs. integers: http://milesmathis.com/cant.html

    You claim the “zooming in” feature is just a geometric trick to aid students, but it is they way the calculus proofs are described. Just check Wikipedia: “Geometrically, the derivative is the slope of the tangent line to the graph of f at a. The tangent line is a limit of secant lines just as the derivative is a limit of difference quotients. For this reason, the derivative is sometimes called the slope of the function f.”

    And, yes, physicists are wrong in assuming that point masses exist, that’s why equations start to blow up in their faces and that is why they have to assume that photons have no mass, otherwise their math falls apart. A mathematical point on a graph is a physical interval.

    Miles has actually quite some corrections, calculations where current physics has no theory and predictions on his website that are sometimes more accurate than current physics but 1) you have’nt read that and 2) current physics has already an ad-hoc theory, 3) they choose to ignore or hide a problem.

    As an example take Miles’ calculation of the axial tilt of the planets and his explanation and correction of Bode’s law or his calculation of the magnetopauses of Earth and Venus. Current theory claim it is all coincidence while Miles can show it is a direct outcome of his unified field that was already found by Newton and Coulomb but incorrectly interpreted as two seperate fields by them.

    His articles are already presented as small digestible chunks, however it might require to trace back some articles to read about basics which could be against your convictions.

    • Steven says:

      Great detective work…

      I have no intent to read Dingle since the book on SR has been closed and the physics community has decided to move forward (like always) despite some obviously glaring misconceptions in Einstein’s derivations.

      It is also clear that Dr. Sfarti has’nt bothered to read Miles Mathis, he just put him on his reference list.

      Mathis arguments are that SR is just the Doppler effect applied to photons and that Einsteins conceptual derivation describes a second order Doppler since in his diagrams two velocities are involved. Correcting the γ formula for those facts leads to not negligible correction. About 4% for instance in the gravity field of the sun. That explains a.o. why Einsteins starlight bending number was corrected later from 1.68 to 1.75, why Saturn has an anomalous precession according to GR and also why recently the proton was found to be 4% smaller than expected.

      If you are up for a discussion about this be my guest.

  13. Dan says:

    I gotta ask the the guy above that has a problem with 1/2 being less than 1. If you deny the existence of 1/2 you must deny the existence of all other non integer rational numbers, or completely abandon the concept of multiplication. DO you have a problem with all rationals, or just rationals less than 1?

    Even further to this have you ever heard of Hilberts school of thought on formalism, one doesn’t have to affix “real” meaning to mathematical constructs, they are just interpreted as marks on a piece of paper with certain rules of inference. 1 + 1 = 2 does not have to be interpreted in terms of counting. Just marks on a paper that are devoid of meaning, it just happens that via our minds we can use it as a tool to understand understand our surroundings.

  14. Dan says:

    I just figured it all out. Mathis isn’t a crank! He is one of two things, maybe both. I noticed he has a donation thing on his website to support his ‘research’. So perhaps these outlandish claims are made to fleece money out of idiots that believe him. The other possibility is that he writes this up for laughs. Specifically he makes up something incredible like the derivative of ln(x) is horribly wrong, then laughs at all the idiots that buy his garbage or supposed book, and laugh even harder as people like the blog owner and myself that spend their time trying to explain his errors to others. It’s like he gains happiness by wasting others time. Maybe it’s both. One of these options must be the truth, there is no other reason a rational person would create such slop. Although perhaps I’m wrong and rational people much like rational numbers less than 1 do not exist :)

  15. Dan says:

    Looking at other nonsense put forth by Miles Mathis I came across http://www.fqxi.org/community/forum/topic/595. There is poster there ‘Steven Oostdijk’. If you read one of Mathis’s papers you see he overuses the word ‘strawman’. Now on the above web page you see Steven overusing the same word, and his language and style are almost identical to Mathis’s. This leads me to believe the posters above; ‘steven’, ‘oostdijk’, and mathis are one and the same. Apparently instead of defending his own nonsense he creates virtual identities and backs himself up using them in an attempt to mimic some perverted sense of peer review.

    If anyone is interested why I’ve read anything he has written, there is a simple explanation. An undergrad at my school was given an assignment to select a math crank, choose one of his ‘papers’, and discredit the claims in it. When he asked for my help we came across MM’s webpage. I should also mention the ‘paper’ we chose to discredit was the one where mathis claims to find a problem with differentiation of the natural logarithm, and then go on to ‘fix’ the problem. In this horrible ‘paper’, he confuses the ‘d’ in the differential operator with a variable (and no there was NO mention of differential forms that might explain the confusion), for some reason thinks the true slope of a tangent line is the average of the slopes [f(x+1) - f(x)]/1 and [f(x-1) - f(x)]/1, that limits involving ‘d’ (supposed to be h) cannot be taken as ‘d’->0 only as ‘d’->1, and a host of other absolutely awesome crankness.
    Looking for a flaw in another’s work is an essential part of the peer review process and can only serve to benefit the entire community. However, one guy claiming to find fault, and then provide his own ‘correct solution’ to over 300 years of modern math and physics is megalomania at its finest. Even were one to assume that there was some fault (which there is not) with the calculus, non-Euclidean geometry, QED, GR, or ODEs, either someone else would have found it and won a fields medal, or internal contradictions would be everywhere, or some advance race of super duper aliens would enslave us as punishment for our ignorance!

    This guy is one of the internet’s most prolific cranks. The more people with a real knowledge of maths and sciences indulge him by arguing with any of his various aliases, the greater the soapbox we are building for him to preach from. The more opportunity he is given to spread is ‘research’ the greater the probability that a young impressionable mind will be perverted, and that is his greatest threat. He should be ignored and forgotten as nothing he has done is even worth a passing mention or a footnote in the history of science. Perhaps if he stops receiving any attention, even in the form of mockery, he will just go away.

    Those are my two cents.

  16. Steven Oostdijk says:

    You guys are too busy chasing ghosts. I do not even live in the same part of the world as Mathis.

    Just provide him some real arguments against his works instead of parroting textbooks and he’ll be happy to conclude being wrong. I only found out he is’nt.

    A circle is as poor an approximation of a curve as a straight line is, that is the only thing you need to understand.

    • D says:

      Steven, instead of calling us parrots please address my argument about formalism, it is contained in my first post in this thread.

      In addition to this I ask you to address the fact that according to you and mathis taking a limit a t->0 is a nonsensical, ill defined, and illegal operation. However in his short paper on Pi = 4 he does this implicitly, and in the full paper on Pi = 4 he does this explicitly. This is an internal contradiction which now implies one of the two proofs is not consistent within his framework.

      Now I ask you or Mathis to address the specifics of my above paragraph and answer why he feels it is ok to take the limit as t->0 in one case but not the other. If either of you want to be taken seriously you will address my points in a clear concise manner without resorting to asserting that I am parroting a textbook. I have made no mention of anything outside of Mathis’ own work and posed my question entirely within his framework.

      I look forward to either of your rsponses.

      • Steven Oostdijk says:

        I’m sorry to trigger your nerves.

        Let’s get back to your remarks about formalism: “I gotta ask the the guy above that has a problem with 1/2 being less than 1. If you deny the existence of 1/2 you must deny the existence of all other non integer rational numbers, or completely abandon the concept of multiplication. DO you have a problem with all rationals, or just rationals less than 1?”

        I had no problem with the number 1/2 (I would with 0.5), my argument was that 1/2 is relatively speaking as far away from 0 as 2 is from infinity. You can only argue 1/2 is “smaller” if you equate a universe of two elements equal to a universe of one element. I think that is an oversimplification.

        “Even further to this have you ever heard of Hilberts school of thought on formalism, one doesn’t have to affix “real” meaning to mathematical constructs, they are just interpreted as marks on a piece of paper with certain rules of inference. 1 + 1 = 2 does not have to be interpreted in terms of counting. Just marks on a paper that are devoid of meaning, it just happens that via our minds we can use it as a tool to understand understand our surroundings.”

        For me that is as useful in math theory as fairy tales in literature. Great for kids, but one day you need to teach them about the real world.

        “In addition to this I ask you to address the fact that according to you and mathis taking a limit a t->0 is a nonsensical, ill defined, and illegal operation. However in his short paper on Pi = 4 he does this implicitly, and in the full paper on Pi = 4 he does this explicitly. This is an internal contradiction which now implies one of the two proofs is not consistent within his framework.”

        Could you show me where? I’m travelling at the moment and not willing/able to spend much time studying. One cannot go to 0 in a ratio, but otherwise it could be valid.

    • D says:

      Oh yeah I forgot, what does a circle approximating a curve vs a straight line approximating a curve have do with anything I said?

      That being asked, I again request you address my above post and not let the conversation change to a debate of best local approximation of a curve. If you wish we can debate that afterwards.

  17. D says:

    Steven,

    1/2 is not as far from 0 as 2 is from infinity. The reals are a metric space with the Euclidean metric d(x,y) = |x-y|. Now d(1/2,0) = 1/2 and d(1/2,infinity) is not a well defined quantity as infinity is not a number. So 2 is not as far from infinity as 1/2 is from zero. Further your notion of ‘relatively far’ is not a well defined concept as currently presented. Please make this precise. State something like : given two pairs of real numbers a,b and c,d, a is “relatively” as far from b  as c is from d if . Why does 1/2 being smaller than 1 require equating a universe with one element to a universe with two elements? I am asking for you or Mathis to provide me with an explicit contradiction within the axiomatization of the rationals that is caused by 1/2 < 1. I have countered your arguments with very precise statements and posed very precise questions. I am well versed in the material, so please don't feel that I somehow require excess verbiage to understand you. If at all posibble make  precise statements as I have and formulate then I'm the symbolic language of real numbers  or metric spaces. I will point out that if statements cannot be made precise in this language then the statements are outside the scope of the real number system or metric spaces. If you like you can extend the rationals to the reals via limits of Cauchy sequences, this is easier to work with than Dedekind cuts, although I am ok with cuts if you prefer.   

    The formalism school of thought is not meant to be a tool for math theory it is a just a school of thought. Don't you think comparing to fairy tales is an oversimplification? That is too easy a way to dismiss my point without actually addressing it. I could just as easily dismiss anything you say as crankness, however that would not be fair to your ideas as it does not critically address them. Please don't dismiss my arguments without addressing them in a specific way.

    What issue do you have with the notation .5? It is a perfectly clear unambiguous notation to describe the number 1/2. 

    Why exactly can one not go to zero in a ratio. What axiom Forbids it? What internal contradiction does it cause? How is it not a well defined concept?  

    Now I have a suggestion to make. Since you two believe there are flaws with the axiomatization of the real number system, why not propose your own axiom set and allow us to examine it? I'm sure people here will examine it, if it is inconsistent someone will prove it, if it is equivalent to the current accepted axioms someone will prove it, and finally if it is consistent and offers some advantage over the current method please demonstrate they are not equivalent and illustrate an example of it's advantage. Further, if nobody here can find any fault in it, we will admit it. 

    A very large part of getting a method/ theory/proof accepted is standing up to peer review. If your theories are correct and you want them to be accepted, then you must be prepared to defend them in a rigorous way with clear explanations and no handwaving.

  18. D says:

    The above shoul read:

    State something like : given two pairs of real numbers a,b and c,d, a is “relatively” as far from b  as c is from d if some condition is true.

  19. D says:

    I would like to make an additional point. You stated:

    “I had no problem with the number 1/2 (I would with 0.5), my argument was that 1/2 is relatively speaking as far away from 0 as 2 is from infinity. You can only argue 1/2 is “smaller” if you equate a universe of two elements equal to a universe of one element. I think that is an oversimplification.”

    I would like to counter in a more rigorous way. The real numbers with the metric d(x,y) = |x – y| are a metric space. If anyone doubts that, well, there isn’t much to talk about then. d(x,y) is a distance function for those that don’t know.

    So I would like to define “smaller” in a rigorous way. I propose x is “smaller” than y if d(x,0) – d(y,0) < 0. Now I think we can all agree that 2 is equivalent to 4/2, if not, then again we have nothing to talk about. So

    d(1/2,0) – d(4/2,0) = |1/2 – 0| – |4/2 – 0|
    = -3/2 < 0. Thus 1/2 is smaller than 2.

    I fail to see this universe of one vs universe of two elements point. I also fail to see the distinction between 1/2 and .5. Neither one of them *is* the number, they are simply notation for the number. There is no reason to think an object cannot have two dfferent symbols that represent it. I think you confuse the notation for the object.

    I beleive that settles that.

    Some time has elapsed since my last post where I asked for specific non-handwaving responses to my very specific questions or arguments. There has been no response to my critique, and certainly no defense against it. Thus I submit to all who read this that the case is closed and the opposition acknowledges that I, and most everyone else here, are in fact right.

    In closing I would like to say that reading anything with a critical mind is an admirable quality. How would we ever correct our mistakes if our writing was never questioned in the first place. Were it not for critical thinking, well, then the planets would still orbit the earth, the Aether would still exist, and Einstein would never have given us relativity. Finding an error in someones work is not a bad thing, it brings correctable mistakes to the authors attention and ultimately they are corrected, and it also brings uncorrectable mistakes to light so that the conclusions drawn from them may be reconsidered. On balance, the peer review system has fostered rather than retarded scientific inquiry. However, for one man to find fault with the almost all major ideas and theories in the past 400 years of math/physics, then to go on to "correct" or "fix" all them all, well that is megalomania at it's finest.

    To be sure, following your ideas when they may go against the grain of conventional knowledge is a brave act, even admirable, for you may be on the path to scientific revolution. However when your ideas run counter to the all the basic ideas of mathematics and physics of the past two thousand years, you must ask yourself one question: 'what is more likely, that everyone for the past 400 years was wrong about not one, but almost all developments, or do I simply fail to understand the fundamental concepts'.

    I leave it to you to decide which is the case for one Miles 'infallible' Mathis.

  20. lily says:

    Charles,

    You might already be aware of this but Underwood Dudley has written several books on the subject. I’ve only read two of them but they were both pretty entertaining and had a large number of examples.

    http://en.wikipedia.org/wiki/Underwood_Dudley

  21. Jonathon Hook says:

    Well it seems some are in disagreement with Miles mathis, and to be expected. Even if some, all, or none of Mathis’ ideas are correct, or marginally so, a greater point is being missed. The same point is missed in virtually all academic settings. The fact that everything (including the theories proposed by notably intelligent individuals) should be in question at all times, unfortunately it is not taught to others this way. There should at no point in time be an idea (theory, law, etc…) that is beyond question or beyond refute, and to simply claim that Mathis is boldly contradicting minds of higher caliber and therefore he is wrong, or I found an error in his logic therefore it must be wrong, would be false, and would have resulted in things like Einsteins Theory of Relativity being completely discarded once the errors were found long ago. His type of thinking should be encouraged in all levels and types of learning and education. Even if someone spends a great deal of time trying to refute a theory only to find they could not, if nothing else, they now have a thorough understanding of something which may be a cornerstone of their entire undertanding of a topic.

  22. wh says:

    @D,

    I’ve looked into Miles Mathis’ texts and I found that most of it don’t deserve any attention at all, as far as science goes. On the other hand it’s commendable that he feels the freedom to look into issues which are usually taboo in a scientific discourse and present his opinions without hiding behind a pseudonim. In doing so, it appears there expressed some things which deserve further analysis to avoid leaving untied ends. As far as I can see, you have undertaken the task to bring in rigor in this conversation so I’d be curious to hear your comments regarding this: http://milesmathis.com/calcor.html.

    As in most of his writings Miles Mathis is concerned with the physical meaning of math (although most of his physical considerations are unacceptable), not with the math per se. Thus, in the text of the above link he is not disputing that the time derivative of \( x(t)^2 \) is \( 2 x(t) \frac {d x(t)} {d t} \) but is strongly opposed to Lagrange’s assumption that \( \frac {d x(t)} {d t} \) is velocity \( v \). He argues that if that’s indeed velocity then the said time derivative should also be written as \( 2 x^3 / t \) which it is not. I’d like to hear your arguments regarding this objection by Miles Mathis because it concerns the virial theorem in a fundamental way. It would be nice if you could illustrate your answer by a wolframalpha graph.

  23. wh says:

    Sorry, don’t know how to type LaTeX code in this site and couldn’t find instruction to do so.

    Here’s one more try: $$ \frac {d x(t)} {d t} $$.

  24. wh says:

    @All,

    I posted a comment here http://actascientiae.org/blog/?p=65#comment-5 just because latex provides more clarity.

  25. RockWalls says:

    @All

    I do not understand the denial of the Mathis fellow. Seems to me it is like the argument about which meat is better in a burger Black Angus or BeefMaster. Nobody or a very few could tell the difference. But, I guess the positions must be formed and defended.

    Please tell me why the universe cannot be discrete. A TV screen is discrete doubly so now as both the signals and pixels are discrete. So, why not the universe at a high enough frequency and how high would that be? Is there a limit? Beyond the Planck lenght and its associated time.

    Thankyou all. Please be kind and sweet as I am not a scientist or scholar like you but, I do work in the industry that provides our food.

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