Why is Math so Powerful?

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To be frank, I find you very hard to follow at times. This is quite common when talking with the insane, as their thinking is too warped to share the same frame of reference. It's also quite common when talking to those with communication difficulties, including those for whom English is not their first language, and it's something of a stretch to call these insane. Since I don't know you well enough to make any assumptions, I can only state that I struggle to understand you at times - please try to keep things clear and simple wherever possible.

Let's start here. What do you mean by "true believers"? I've already stated that the concept of believers in the religious sense is nonsensical. What do they actually believe? That mathematics is the mind of God? That everything can be expressed mathematically? That will and fate are determined by mathematical laws?



I didn't follow this at all.



Or this. Surely "Casper the Friendly Ghost exists" is a premise or conclusion, not an axiom? Why does not seeing him necessitate his being invisible? I can't see China, but I've had reports that it is very visible.



Plain English is indeed a problem. Definitions are needed as we go.
You seem to be stating that conceptual things exist as physical phenomena. This is something that you'll find very little support for. I am reminded of the tale of the tourist in Oxford, who was shown all the Colleges and asked where the University is. We have a number of concepts that do not actually exist - money is a good example. Or transsubstantiation - the idea that the bread and wine that Catholics take in Mass turns into the blood and body of Christ. No physical change occurs. And what about more esoteric concepts, such as beauty or mercy? How do you measure these things, for surely anything with physical presence can be measured?



Numbers are things, albeit in a conceptual fashion. There is no such thing as two in a real sense. Yet within a conceptual framework, numbers and axioms can be used to prove all manner of things *within that framework*. The wonderful thing is when you map that framework onto reality, it still holds. If I have two cows and you have two cows, we have four cows between us. If I fire a cannonball of mass m with x kilos of gunpowder at an angle of r, it will travel y metres in t seconds - or near enough, because there are more variables involved than that. The more I take into account, the more accurate the result (but the longer it takes).



Your point is meaningless here. The axiom that the angles of a triangle add up to 180 degrees is true within Euclidean geometry, but not necessarily elsewhere. This doesn't mean it is invalid, just that it has limitations (like all truths). I can state "it is dark at midnight" and be entirely correct unless I'm in the arctic circle. That doesn't make the statement entirely false.



The whole point of axioms is that they are true. They cannot usually be proven (they are the basis of proofs, most of the time) but if they prove to be wrong, they are not axioms. But, again, they may not apply to every framework. Spherical geometry has different rules to Euclidean geometry. The rules of chess do not apply to draughts, but we do not disallow castling because of that.



You're losing me again. Pure mathematics is true - but only within mathematics. I can easily draw a "triangle" whose angles don't add up to 180 degrees. In fact, I'd struggle to draw one that did, because straight lines are difficult when drawn freehand. But in a mathematical model, triangles are always true, even if our illustrations of them are wonky.



No, not really. Just a lot more complicated, and we don't know all the rules.



I question whether he turned "religion" into Sociology. He might be insane. I have nothing to go on. Certainly sociology is a very "soft" science with very few axioms and a lot of conjecture!

Except for the trivial parts that compute statistical correlations, sociology is not a science at all. It is a collection of doctrines, hunches and beliefs unable to make quantitative testable hypotheses.

ruveyn

You are trying too hard. There is little use in attempting to find meaning in the ravings of a lunatic.

As far as I can tell, Tadzio appears to reject the validity of formal mathematics- that is, the process of defining axioms and reasoning from those to theorems.

Hi Thom Fuleri,

I have Temporal Lobe Epilepsy and I know my communications are difficult to follow at times. Whenever there is a conflict with my communications, I refer to the records of my communications and the references I used in initiating the content of my communication. A major problem with using these techniques with references is the conflicts over plagiarism and the notion of context.

Maybe you don't have the same access to data-banks on the internet as I do. But, I can get access to other internet connections, and they return nearly identical search results of the phrases, sentences, and paragraphs that I have used (and cited) in books and other resources within very close contexts. I do have the added advantage of having physical copies of most of the books I have used, and to all the books I have directly cited.

As your responses remind me of the aggressive and defensive reactions I received in grade school over Christmas Holiday celebrations that I didn't freely and openly embrace, including with fellow students who simultaneously claimed to be "atheists" and to have knowledge that "Santa Claus" was a real individual living at the North Pole, I realize the anxiety levels created in those individuals whose first line of defense is to label anyone with a different set of observations and beliefs as being "insane".

There are elements of humour in many issues, such as on another forum, ruveyn making the declaration that "Angels of the Lord don't have dongs". Based on a confusion of concepts, which I know are often differently labeled by different sources, here, seems to extend beyond the problems of labeling, as you appear to display the aspects of what I label as a "true-believer" in a set of beliefs that more than frequently returns nonsensical results that are best not to be acted on, or responded to. What I am talking about here is moderately summarized in the book "How to Think About Weird Things" by Schick & Vaughn (2002).

I used the word "rigorous" as in:
"RIGOROUS is nearly akin to RIGID, but is a stronger word, having reference to action or active qualities, as RIGID has to state or chracter; a RIGID rule may be RIGOROUSLY enforced." From: "Funk & Wagnalls Standard Handbook of Synonyms, Antonyms, & Prepositions" by James Fernald (1947), page 386.

Your contention that "physical reality" is "not really" more "rigorous" than works of fiction, reminds me of another of ruveyn's forum postings about "Big Government, Small Brains, Dumb Laws" with the issue of whether the warning on a Superman Costume of "Wearing of this garment does not enable you to fly", was a stupid & needless warning, and ruveyn's response of:
"Not a joke. When I was a kid I lived in the Bronx. I was up on the roof of the 7 story apartment building where I lived (I was flying a kite) and I saw this Puerto Rican kid tie a towel around his neck. He got up on the parapet and jumped off. He thought he could fly like Superman. I heard him crying Oh chit, Oh chit all the way down. There is nothing funny about this. It was very upsetting."

With you quoting my sentence of "Physical reality is more 'rigorous' and more unforgiving than attempted works in that of fiction", with your response of "No, not really. Just a lot more complicated, and we don't know all the rules", I strongly believe that you may be an excellent example to illustrate the need for such warning labels on consumer products, and I'm concerned such warnings may also be needed with any "critical thinking" issues in such instances.

Two of the books I'm referring to at this instant are numbered in the "Oxford Logic Guides" as numbered volumes 19 and 24: "Godel's Incompleteness Theorems" by Raymond Smullyan (1992) and "Introduction to Logic and to the Methodology of Deductive Sciences" by Tarski (1994)(previously cited). It's quite evident that neither you, or "Orwell" with his postings, have made it through the prefaces of any introductory books of such. Certainly, Tarski's contention about the logic of the empirical science of such as Physics existing at all, is much too large of an intellectual challenge to your sanity:
"I must say that I am inclined to doubt whether any special 'logic of empirical sciences', as opposed to logic in general, or, to the 'logic of deductive sciences', exists at all (at least so long as the word 'LOGIC' is used as in the present book - that is to say, as the name of a discipline which analyzes the meaning of the concepts common to all sciences, and establishes general laws governing these concepts). But this is a terminological rather than a factual problem." page xii, ibid, Tarski.

I am trully sorry that I carelessly assumed that you and "Orwell" might have made it to page 1. The entertainment of the concept of "hopeless" as to understanding, seems to also extends such across to the very much the same concept in the sentence:
"Hence Putnam cannot embrace realism without embracing classical logic, and hence his argument to endorse quantum logic because of realism about quanta is a hopeless case." http://en.wikipedia.org/wiki/Is_logic_empirical%3F

But here, I doubt if you will ever figure out why five quarts of water with then an added quart of salt mixed in, will generally give you less than six quarts of solution, since every "sane" person "knows" that 5 quarts, plus one quart, equals 6 quarts.

Tadzio

They don't believe that what is called mathematical induction is different from what is called empirical induction. They know that what they mean by mathematical induction and what people mean by empirical induction are different. What people believe about them and what people call them are irrelevant.


There may be sets of axioms which could be considered closed for some purpose or other, but that is not a restriction against coming up with new axioms or sets of axioms.


Empirical induction is not mathematical induction. By empirical induction we can come to the conclusion that someone whose heart is beating will continue to beat, which is statistically pretty close, since any given heartbeat is unlikely to be anyone's last heartbeat.

By mathematical induction, we would not reach that conclusion (even approximately) in the first place. True, you could show that your heart is beating now. But there is no way to show that because it is beating now, it will continue to beat.


Wrong. As you noted, it is a cycle that is not a perfect ratio.

Also, given the other things that could perturb the Earth, Moon, and Sun, even if the ratio was perfect, we would not be able to construct a mathematical proof by induction, since given that it works now, we can't draw the conclusion that it will continue to work.


If the square root of 2 is a ratio between 2 whole numbers, let's see if we can find out what it is. Call the two numbers a and b, with a on top, and since we can always write a ratio in terms of numbers without common factors, let's say that a/b is a ratio without common factors. So (a^2/b^2) = 2. If we multiply by b twice, we get a^2 = 2*(b^2). So a^2 is even, so a is even. Since a is even, a/2 is a whole number. Let's call that number c. So a^2 = (2c)^2 = 4*c^2 = 2*b^2. If we divide by 2, we get 2*c^2 = b^2, so b^2 is even, so b is even.

But if a and b are both even then they can't make a ratio without common factors, so the ratio of a/b doesn't make sense if a and b are both whole numbers.

So no, it's not a fable.


Skinner was a psychologist, right? What does this have to do with mathematics?


This sounds like you might be trying to criticize set theory, but you haven't been specific.

This does not disprove mathematics.


I felt the impulse to give you a great deal of leeway due to the temporal lobe epilepsy, until I read this.


If this is what you think pure mathematics is, stop calling it pure mathematics. That isn't it.

The quotation marks around the quote from Tarski, means Tarski wrote this paragraph. I know that WP places "Tadzio wrote:" as a routine "quote" process, but your intelligence level may not allow you to understand that, and I don't want any responsibility for your erroneous conclusions. Copyrights prevent me from posting the entire book here, so I'm sure you'll have to learn what a book is before you'll understand what a quotation from a book means.

I suggest you read a book on applying Logic and Mathematics to instances in everyday life, instead of searching for "BigFoot" in your axiomatic daydreams.

Take your prejudicial impulses and try to improve your own state of intelligence, as I am already utilizing reasonable accommodations that must have also given me more than an unfair advantage in forum discussions.

Tadzio

Hi ruveyn,

Using a book no longer under copyright, with citations that still are (so not directly quoted or listed):

Pure Mathematcs is not empirical, but Applied Mathematics is empirical.

Deductions are formed from inductions from a number of collated instances, through common attributes,
to a general principle, which forms the basis of a conclusion as the premise of a new deduction. The highly
repeated and refined processes of these principles lead to the most useful premises, and are frequently
taken as axioms.

We do physics with Applied Mathematics. We do not do physics with Pure Mathematics.

Mathematics can be a powerful tool, but in the non-methaphorical senses of the word "tool", only Applied
Mathematics is a powerful tool.

In the metaphorical sense of the word "tool", Pure Mathematics can be a powerful tool, but it can be a tool
of an intellectual roadblock.

The book "Scientific Theism versus Materialism: the space-time potential" By Arvid Reuterdahl, and available
for free at books-dot-google, page 45 for a search of "'pure mathematics' ether", reminded me of a book
written by Einstein, and how Einstein had to combat all the true-believers in the "ether" that light propagated
through by the principles of Pure Mathematics. Einstein called this "ether" nonsense, and declared that the
physics of light was not confined to ridiculous concepts justified only by "rigorous methods of mathematics".
Einstein finally refused to even spell the word "ether", and he started using that e**** word when he wanted to
disparage the nonsense.

With Sociology, the nonsense of "Scientific Theism" sounds very appropriate, much like a "Christian Scientist"
along the lines of Mary Baker Eddy.

Here's the internet location, which often ruins the browser margins on WP:
http://books.google.com/books?id=vbVCAA ... er&f=false

Other websites have bits of info:

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

http://en.wikipedia.org/wiki/Is_logic_empirical%3F

and plenty more, but only beliefs of "true-believers" are protected from attack here at WP under "Computers,
Math, Science, and Technology", so please forgive my being an infidel & heretic.
(I didn't put the word "fictional" in front of the word "ether", because few people here can differentiate
between people who insult the belief in "Casper the friendly Ghost" and people who are true-believers.

Tadzio

Ah, that explains the style. One suggestion that might help - write less. It is a common misconception that adding more text to a point will make it clearer (in the same sense as adding more red to your paint will make it redder) but really the opposite is true - if any of it is unclear or seems contradictory, it confuses the point further.

On the plus side, you have a clear understanding of paragraphs, which puts you well ahead of at least 80% of internet posters, and you can spell, which puts you in the top 5-10%.



The issue is not usually that I don't believe your source material (I can look these up if I need to verify them) but that it doesn't seem to support the point you are making.



My responses seem aggressive? These are positively friendly compared to some I've had on other forums! There is a difference when debating between aggressive posts and merely spirited ones - if the post is an ad hominem attack on you rather than the ideas you're posting, it's aggressive. Someone disagreeing with your beliefs and ideas is a good thing - it's what debate is all about.

As for insanity - I'm of the opinion everyone is insane (especially me). How else do we cope in a world that has no meaning, will eventually lead to our own death and in which entropy and decay are the only constants? If we truly comprehended the fundamental pointlessness of existence, we'd never do anything again. The interesting part is how insane we are, and in which direction.



Heh. Puts me in mind of something one of my school friends used to say - "nothing is foolproof to a sufficiently qualified fool." Basically, some people are idiots, but that doesn't prevent them also being resourceful or intelligent. Government is a good example!

Perhaps we should quantify "works of fiction", however. Mathematical axioms are not "fiction". They are entirely factual. They merely relate to a conceptual world that is based on our own, but simplified. Mathematical proofs are very rigorous, but *in this conceptual world*. They still apply in reality, but not accurately - this is why it's been "proven" that it's impossible for bees to fly, kangaroos to jump and men to run a mile in less than four minutes. These are all true with the mathematical models used, but the models were wrong. "Garbage in, garbage out," as they say in computer science.



Now THAT is bordering on an ad hominem. No, I have not read those books. This doesn't prevent me from having an understanding of the concepts.



I think you have misunderstood Mr Tarski. He states there is no *special* logic for emperical science, not that there is no logic at all. That is, logic applies to empirical and deductive science in the same fashion. And this is the whole point - mathematical logic is the same as empirical logic. But they have different frames of reference.



Don't get started on quantum mechanics. I'm not sure anyone understands that!



People still use quarts? In any case, it's the wrong measure to use. Now, if you were using kilograms, that should hold, because that's measuring the mass.

Still, using volume measures is a good example of how the wrong models can result in wrong results. Here the model used is that water and salt are solid - that they completely fill the volume given. In these circumstances, five quarts of one and one quart of the other would indeed give six quarts. However, with a little knowledge of physics, we know that this is not the case - that both contain "gaps", and that salt dissolves in water in such a way that it fills some of those gaps. You don't even need to know that much - the same quantity of water takes up more volume when frozen and a LOT more volume when boiled. Even without the salt, we can understand that five quarts of water could really be any amount of water, depending on temperature and pressure.

Science applies experiment to knowledge. A quart of salt in five quarts of water does not produce six quarts, so the deductive model of the empirical experiment is wrong. We don't throw out logic. We adjust the models until the experiment works.

Greetings Any Hylephobiacs:

"Cosmos" by Carl Sagan (1980, 1985):
"In love with whole numbers, the Phythagoreans believed all things could be derived from them,
certainly all other numbers.....[they] suppressed the knowledge of ..[square root of 2]... and
the dodecahedron.....Plato urged astronomers to think about the heavens, but not to waste
their time observing them....A Phythagorean named Hippasus published the secret of the 'sphere with
twelve pentagons,' the dodecahedron. When he later died in a shipwreck, we are told, his
fellow Pythagoreans remarked on the justice of the punishment. His book has not survived."
pages 151-152.



"...Its analogue we may call hylephobia, or morbid fear of materialism, also a very modern distemper, which
afflicts, now and then, a philosopher with a horror of contact with the fresh facts of science so necessary to his
survival in the world of modern thought, and impels him to try to purge every element of matter from facts he
cannot escape. Hylephobia, however, is now often regarded as a sacred madness, as epilepsy used to be. It befalls
only the good; and the richer and fairer the world of sense, and the more violent the phobia against it, the more
surpassingly rich and fair and real must the purely subjective, rational, ideal world appear. All the wisdom of
scientific psychology melted in this author's [Dr. Borden Parker Bowne's] crucible is but slag and dross,...."

Tadzio

I am bemused. Are you suggesting that pure mathematics does not exist, and that all science should instead rely on guesswork/superstition?

Coming up with a hypothesis (be it successful or not) has an element of guess work. Hypothesis do not flow from facts. Given any finite set of facts there are in infinite number of hypotheses that can account for them. Hypothesis must be -invented-. Hypothesis and theories do not leap out of collections fact like frogs leap from lily pads.

ruveyn

I'm sorry to upset any hylephobics out there, but there's only two options.

1) The universe is ordered, and things happen consistently.
2) The universe is random, and consistent events are purely coincidental.

The first case is a materialistic universe, and all science hinges on this being true. Indeed, so does any philosophy of existence - Christianity makes some very big assumptions that God exists, created everything and is basically a chap with a purpose (albeit one unknown to us) - it wouldn't work if tomorrow God could be a stick of asparagus conducting symphonies on Pluto, or that the universe can be four billion years old today and fifteen years old next week. There has to be a consistent narrative.

There is no in between - the universe is entirely ordered, or it is not ordered. Any random element will throw the whole of existence into doubt. Indeed, all structure could suddenly vanish without cause.

If the universe is ordered, we can spot the patterns. That is what science is all about. If it isn't ordered, science is meaningless - but so is everything else. It therefore makes sense to us to assume it is ordered, because it won't matter if we're wrong but we'll make progress if we're right.

There is something so incredibly flawed with that premise that it boggles my mind how you came to that conclusion. It's as if certain basic concepts such as cardinality and limits were lost on you when you wrote that statement. Rather that go into detail on the additional possibilities let's simply point out your biggest flaw.

The one HUGE flaw in your design( And in the designs of many in the fields of philosophy and the sciences; so its not only you who errs with such arguments), is that a narrative must remain absolutely and in all ways consistent, in order for you to gain or apply knowledge. There is a basic principle being overlooked.

The only way for this argument you presented to have any possibility whatsoever to be true, is if we are dealing with a completely STATIC universe.(And given what we do know about the universe I am wondering why you would in any way believe that to be the case.) If however, we are living in a DYNAMIC universe, than what the universe is, and what is capable of, becomes exceedingly more complex than that for which you give it credit. This one concept of a DYNAMIC system, eradicates any segment you have of being correct, unless you can prove that the universe is truly and wholly STATIC for all time(something I believe you or anyone else for that matter will have trouble doing).

By not perceiving the idea of a DYNAMIC vs STATIC universe you are making many grave assumptions.

Irregardless, your argument for an ordered universe has many holes in the proof and disregards much.

What?!

The universe is and isn't static, depending on how you look at it. From our viewpoint it IS dynamic - it constantly changes, which is kinda important for us. I wouldn't be able to type these words if it didn't. You've clearly not understood a fundamental concept in my earlier post - the universe is not constant, but consistent. That is, if everything is set up the same way, it will have the same result.

If you look at time as a dimension of the universe, however, it IS static - you can't change what has happened, and you can't change what will happen, as it hasn't happened yet. Whatever we shape the future to be will be the future.

Basically, the universe can change - but only within static conditions (which we label as the laws of physics).

Not true. The laws of physics are not deterministic. That is the main point of quantum physics.

The "block universe" is not a faithful model of reality.

ruveyn