Why is Math so Powerful?

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Mathematical induction isn't that "difficult to grasp" really.

All you do is start somewhere and show that you can always get to the next step.

Consider a ladder. There is a first step on the ladder. Prove you can get on the first step.

Then prove that, if you are already on a step on the ladder, you can step up one step and get to the next step.

Since we can get on the first step, we can get on the second. Since we can get on the second step, we can get on the third. And so on forever.

The other kind of induction is called empirical induction or Baconian induction (after Francis Bacon).

I have seen a lot of white swans and that is the only kind of swan I have seen, therefore all swans are white. Actually they are not all white. Black swans have been seen, rarely.

ruveyn

ruveyn

Beauty is subjective. Using modifiers and/or synonyms to modify expression or descriptions of "beauty" does not make "beauty" less subjective (maybe it makes beauty more verbose at most). Fractals are inductive, trying to apply them to "things" (to me the simplest example would be a "coastline"), the "fractional dimension" conceptualized by induction might be applied to describe an attribute of the "thing" by using deduction from the original induction.

Mathematics is not needed to produce music (arithmetic is not needed to pound on a coconut). Introducing "symphony" is somewhat like sneaking in another God into mathematics by using the trap-door of "music". Information theory versus God is in the forum at: http://www.wrongplanet.net/postxf175198-0-15.html

Niccolo Tartaglia (1499-1557) is a historical and World famous mathematician and engineer. He was one of the first in the Western World to use mathematics as a tool for applied mechanics. He is usually included in any address of "Great Feuds In Mathematics". He was a "Galileo" before "Galileo" (Galileo made the point the Moon is not a perfect sphere, because craters were evident on it, while the Vatican responded that it was a perfect sphere, even if they had to coat it with "invisible crystal" to justify it as "perfect", which Galileo said there then were invisible craters on the invisible crystal, not "amusing" the Vatican; "Perfect Spheres" justified a God in the "works" of everything, and now such nonsense is re-infecting science (like "something we can't do without" in physics)).

Tartaglia is also very famous for giving a good kick to the "abstract divinity" of mathematics, with the solution to a vast set of problems that violated "don't think out of the divine box" of "perfect mathematics". Search books-dot-google for "Nicolò Tartaglia 17 horses" to get results like: "In Italy the astute Cardinal Bellarmine, and more fervently various Dominican theologians, had always wanted to keep the New Science penned up behind its own wall of mathematical reasoning, as if mathematics were its only reality." from: "Time, Space, and Motion In the Age of Shakespeare" by Angus Fletcher (2007), page 33.

I was exposed to the "17 horses" riddle in math classes in grade school, and it still is regarded as one of the "Famous Puzzles of Great Mathematiciams" (book by Miodrag Petkovic (2009), Division of 17 horses, Problem 2.11, page 24; and hundreds of other "math" books, but this book also mentions "propriety" in mathematics, as if the infection of Gods is about to erupt again like a herpes infection). Did the "manufactured solution" here offend a "God of Purity" in Mathematics?

Tadzio

P.S.: The wording in grade school included a passerby with a horse who lent his horse to the brothers so they then had 18 horses. The first brother received half the horses, or 9 horses; the second brother received a third of the horses, or 6 horses; the third brother received a ninth of the horses or 2 horses. The passerby then took his horse and left. The arithmetic is really difficult: 9 + 6 + 2 = 17 horses, but this "violates the divinity" of mathematics held by individuals who dislike "the new science",
and in "fact", the passerby must have been Satan!! !.

So the Mona Lisa (as an example - beauty IS subjective) would be just as beautiful as a blank canvas without the need to modify it with all that paint?



Sigh. Fractals are conceptual. It would be a very strange coastline that was entirely fractal. The coastline example simply shows that a shape can have infinite perimeter but finite area, which is pretty weird. Are you going to take umbrage at Hilbert's Hotel next?



A knowledge of mathematics is not needed, just as you don't need to be an expert in biology to get pregnant. Mathematics is not simply needed to make music. It IS music. Music without mathematics is just random noise.



You really don't understand mathematics OR music, do you? A symphony is, in practical terms, nothing more than music with a large band. Instead of a four-part combo, it's an 80-part combo. The numbers are more complicated but they're still numbers.



Someone once said that mathematics is proof that God exists because it is consistent, and proof that the Devil exists because we cannot prove it.

Mathematics is pure. It doesn't map to reality quite so well, because a mathematical model is just a model and is only as good as we make it. It is impossible to model anything precisely, but if our models are close enough we can make limited predictions. There are no Gods in mathematics - they'd need to be proven first.

The problem given is arbitrary, and not mathematical beyond basic arithmetic. It's really a lateral thinking problem. In that light I like it. A mathematical solution would involve fractions of horse, which is leading to a philosophical problem - how many is one horse? No two horses are alike. Indeed, with time and the right circumstances, two horses can easily become three.

Math is unlike any other scientific discipline in that it can be expressed in terms of certainty. And for that reason, it is beautiful. The idea I can describe the world around me in terms anyone can understand is unimaginably powerful. And I've found that once you dive into math as a passion, the more you want to know.

Calling mathematical induction a form of mathematical deduction isn't exactly what I'm talking about. But, I am trying to talk more about mathematics not being closed in a hermetically sealed box, isolated from everything except its own "pure" axioms, though "deduction" as the part of being isolated to axioms as the set to be used to deduce conclusions "from", is different from the rules of logic grouped as used with "deduction" versus some of the same grouped with "induction" as the "axiom of induction" 4(if this axiom of induction is the same as the usage of "deduction", why have the extra & unneeded axiom, no matter the name label?). So, the axiom of induction is also used with deduction, which results in confusion as if both are in absolute contrast with the other. This is a different confusion than the confusion of non-demonstrative induction versus the form of demonstrative induction labeled as mathematical induction.

"This is why it can be (and usually is) misleading to contrast induction with deduction. Induction and deduction are of very different categories. Deduction is concerned with implication and consistency. It is not directly concerned with inference. For a conclusion to be deducible from certain premisses is not for that conclusion to be inferable from those premises. If you believe those premisses and then deduce that conclusion, you learn that your beliefs imply that conclusion. That by itself is not enough for you to be able to be justified in inferring that conclusion. It may instead provide a reason for you to question your belief in the premisses."1

The sealed box view of "not only is every mathematical discipline a deductive theory, but also, conversely, every deductive theory is a mathematical discipline" usually mentions the strong CAVEAT EMPTOR: "It is easy to see that this ideal can never be realized."3 The "to be" sealed box always starts with a contaminating induction from the "outside". This seed from outside the box induction is always from the empirical workings of the world.

Physics, as a science, is inductive reasoning giving us what we know about the empirical world, as distinct from abstract disciplines which have no more existence than a spiritual ghost or that of any of the Gods.2

Tadzio

1"A Companion to Epistemology" by Jonathan Dancy, Ernest Sosa (1992), page 201-204.
2"The Power of Critical Thinking" by Lewis Vaughn (2008), Chapter 8. Ibid. Berlinski.
3"Introduction to Logic and to the Methodology of Deductive Sciences" by Alfred Tarski (1994), Chapter vI "On the Deductive Method", pp.109-112.
4"Fundamentals of Mathematics, Vol. 1: Foundations of Mathematics: The Real Number System and Algebra" by Behnke, Bachmann, Fladt, & Suss (1986), Chapter 1, Pickert & Gorke, pages 94-95.

Look into "Naive Set Theory" for a very large number of different "types" of induction, then check into the paradoxes, then create an infinite set for an infinite different number of types of "inductions". Russell's Paradox is the most famous "lone" example, but the extended Sets of Sets Paradox is more humorous (whether they have Swans of any color or not, "it is necessary also to have at hand a set to whose elements the magic words apply"). As if reality wasn't bad enough, "tradition [also] always conquers pure reason". http://plato.stanford.edu/entries/russell-paradox/ (there's that "viscious circle" again!! !)

http://en.wikipedia.org/wiki/Naive_set_theory (Goodbye Infinity???, then, Goodbye Zero!! !)

Tadzio

Empirical induction has little to do with abstract set theory. It is a way of formulating a universally quantified proposition on the basis of a finite collect of particulars. It is the engine of scientific discovery along with abduction (inferring causes from effects) and analogy/metaphor (waves and particles are metaphors).

The word "induction" is used with two distinct meanings which should not be confused.

ruveyn

You can have sets with elements that are the "empirical inductions", just as if the "empirical inductions" were in a large (infinite???) dictionary of such possible "empirical inductions", and then try to keep the empirical induction regarding such dictionaries from creating a paradox.

Set Theory somewhat destroyed the very old argument that "abstractions" were not "material things", nor "things" at all for use in circular arguments defending the "nature" of the elements compatible with fundamental axioms. Sets with their elements being the simple shadows of ordinary physical objects cast by sunlight is a very simple set that results in intense confusion, namely because shadows do not exist except as an abstraction of an absence of the same level of surrounding level of light "outside" the shadow. Most everyone makes the careless mistake of assuming shadows are somehow materially real, and a majority of people will hold that "a shadow" can move faster than the speed of light, esp. since it can transverse the surface of the Moon faster than the speed of light. Another shadow surprise are sculptures like those of Larry Kagan: http://larrykagansculpture.com/

A "naive" conception of shadows, with set theory applied, allows many other pseudo-miracles, and plenty of blunderous mistakes, much like the way most of the mathematicians were taken in by the Monty Hall Paradox (the APA, their DSM statistics, and cognitive psychs are still making the same old mistakes). The shadow temptation was too strong for some shrinks, and the disasters of "Shadow Syndromes" were born as a minor set of mental disorders, which was/is near total nonsense at a high price.

Tadzio

Tadzio, you seem to be confused by similar sounding words. Mathematical induction is the rigorous deductive process of proving arbitrarily many cases of a claim. Induction in the empirical sense is the leap in reasoning from particular observations to some general claim- eg all swans we have observed are white, therefore all swans are white. Unlike mathematical induction, the form of induction used in the empirical sciences can never give us a 100% guarantee that our conclusions are correct.

We know that's what you were talking about. In putting forward such a claim, you are not just wrong. You are ridiculous. In mathematics, everything proceeds directly from the axioms. This includes counting.

I'm sorry that you find people ridiculous who are not confined to thoughts within your hermetically sealed box of pure axioms.

Presently, I am not a true-believer in Pure Mathematics. I understand that true-believers in Pure Mathematics believe that "Mathematical Induction" is totally distinct and different than "Empirical Induction". I don't actually know whether or not you accept what is commonly called the "Axiom of Induction" and/or whether you regard this axiom as in any way distinct from what is commonly labeled Mathematical Induction processes within your closed set of axioms.

One example of Empirical Induction is that since I remember my heart beating previously, I know through the logic of empirical induction that my heart will beat forever, which is obviously a ridiculous conclusion.

For a more complex example of Empirical Induction I'll try the Metonic Cycle, which has a period of about 19 years, when the "solar year" syncs with a (nearly) common multiple of the "lunar month", and since I have always observed this cycle occurring in history, I therfore know by Empirical Induction that the Metonic Cycle will repeat forever every 19 years. This case of "forever" is also ridiculous, but not to everybody (mainly involving the notion of "forever"). The point of more interest here, is the ratio of whole numbers representing the Metonic Cycle. By the Axiomatics of true-believers of perfect ratios of whole numbers being the mystical cause of the Metonic Cycle, it is not "Empirical Induction" that is being exemplified, it is the purity of "Mathematical Induction" that is being exemplified. This "purity" is best "placed" in with the mystical nature of "The Music of the Spheres", which also includes the numbers often used in "music", and also the "music" previously cited in this very forum as exemplifying "Pure Mathematics".

The lenth of the Metonic Cycle is not a perfect ratio of two whole numbers, so "trouble" soon develops between "true belivers" in perfect ratios and "non-true believers" of perfect ratios of universal absolutes, especially when direct observations record "the problem". As with the fables involving the square-root of two not being the ratio of whole numbers, the axiomatic "incompleteness of the set of whole numbers" to include "numbers" that represent the "concept" of the square-root of two, the problem will be denied, labeled ridiculous, a trouble only with non-believers misunderstanding infinite mathematical inductions through an infinite set of whole numbers, etc., all the way to evil spirits trying to sabotage the perfect harmony of the music of the spheres (I've been around devoutly religious poisonous snake-handlers, true-believers speaking-in-tongues, and people who will practice stoning of "non-believers", so I know that not only are they very unreasonable, they are also very dangerous).

Now, the ridiculuos nonsense of invisible perfect spheres has retreated to the realm of "pure concepts", such as those in axiomatic systems used in "Pure Mathematics", along with nagging paradoxes that have yet to be resolved without creating further paradoxes (one "solution" was the "axiom of induction", which is taken as an axiom to resolve older paradoxes, and is not now, with the status of being an axiom, wholly of any process of "deduction" and/or "induction").

With all of the inductions, I've already cited B.F. Skinner, but before noting mathematics as a behaviour (horses doing mathematics have already been noted too), I'll note Karl Popper and his stance commonly summarized as a doctine of falsifiability, and then I'll note that any paradox of an axiomatic set falsifies the set as being "Pure and Complete". The power of deductive mathematics on the opposite side of the Moon is just about as powerful as a hex-headed screwdriver on the opposite side of the Moon. This is why the Full Moon smirks at Humans who dare to take a peek out-under their cloisters at their self-imposed monasteries of Pure-Thinking.

Now, about the Week having seven days in it.......

Tadzio

P.S.: There's another cursed set of swans: http://en.wikipedia.org/wiki/Falsifiability

You are insane.

I have at times received that response from individuals with much like Mussolini's psyche. It's
often linked to extended practices going far beyond "the usual country knowledge concerning
the planting and pruning of crops according to the moon's phases", and it is a typical Endymion
Complex of being sapped in the tangled situation in a cobweb of moonbeams of failed
self-actualization.

Do you hold as insane everybody that recognizes Easter???

Tadzio

"Nocturne: A Journey In Search of Moonlight" by James Attlee (2011), page 48

There are no "believers" in pure mathematics - at least, not sane ones. The whole point of mathematics is that things can be proven. It's not a matter of belief.



Let's stick with this one. This is empirical induction - the assumption that, if things do not change, they will continue as they are. That's fine. Your heart will keep beating until things change (for instance, death).

A mathematical induction is a more rigorous variant on this and would require two parts - first, that your heart can beat at all (trivial to prove, as we already know it has), and second, that one beat leads to another. If this is always true, your heart will indeed beat forever. It is, however, not always true. Heartbeats are not purely inductive but subject to a billion variables.

He doesn't mean you are insane for your beliefs, but that your beliefs stem from your insanity. Though I do find a certain appeal in classifying religion as a form of mental illness...!

There are "true-believers" in "pure mathematics", but presently, there are not nearly as many as previously, and they are sane, though they at times make very blunderous & stupid mistakes, mainly, IMO, through intellectual inaction, as when they physically "act", they make much the same differentiation that is being erroneously made here, in that their self-contradictory "acts" are not within their definitions of "concepts". (A very simple example is making the argument that you know the axiom of "Casper the Friendly Ghost exists" involves a Ghost that is invisible, not only because you have never empirically seen "him", but also because "he" is non-empirically "axiomatic" (the very old circle of "empirical" versus "non-empirical").

My stance is more of radical Skinnerian Behaviourism, in that "concepts" are not "non-empirical", but are simply subtle, and difficult to observe, empirical physical phenomena. Using "Plain English" is a major problem here, as the usages will be subjected to baseless inductions (as in Chomsky & Rand vs. Skinner).

Your usage of the word "things" would be challenged, as the Pure do not regard "numbers", "concepts" etc. as "things", as they hold them as, maybe, "abstractions", with tons of, IMO, word games. The "word game" is sometimes used that "propositions" are "proven" from the fundamental axioms, but not "proven" in relation to "things" at all. This makes a "true" proposition very limited, with no Pure "use" outside the non-empirical "processes" and the particular set of fundamental axioms.

To simplify with short, careless, examples, use the example that the inside angles of triangles always add to a total of 180 degrees in Euclidian geometry, but in Spherical geometry, the angles total to 180 degrees only in infinitesimal limits, and more than 180 degrees beyond the infinitesimally "small" limits (the "spherical excess" (of Girard)).

With the loaded word "belief", it is a matter of "belief", because you have to believe that the axioms are going to be more useful than troublesome before you try to apply them. Once you make the observation that a set of axioms are not useful more than troublesome, you discard that set, and try another set you construct to be useful that you "believe" might work better. Therefore, when I try to do anything on a large scale on the surface of a sphere, if the scale is large, I will discard Euclidian geometry and use a Spherical geometry and not worry about falling off the edge of the world, though a less than sane Euclidian insurance agent will cancel my life insurance because of the extreme self-imposed risk I take.

An answer might be self-evident to another when the individual takes the "pure" as to mean "true" without violating regard to the limits of the axiomatic set of interest, then complain that "Even if the considerations that the proof involves seem to be PATHOLOGICAL and foreign to the arithmetic spirit that we expect to see in the theory of natural numbers, the end justifies the means" ("Naive Set Theory" by Paul Halmos (1991),pp.46-47), and concludes, as a dunce, that this so reveals beliefs that stem from insanity.

Physical reality is more "rigorous" and more unforgiving than attempted works in that of fiction.

Auguste Comte turned religion into the "science" of Sociology, do you hold him insane?

Tadzio