Apparent Paradoxes

Has science taken a wrong turn? If so, what corrections are needed? Chronicles of scientific misbehavior. The role of heretic-pioneers and forbidden questions in the sciences. Is peer review working? The perverse "consensus of leading scientists." Good public relations versus good science.

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Re: Apparent Paradoxes

Unread postby sketch1946 » Mon Mar 27, 2017 4:30 pm

Augustine from Hippo Regius in North Africa writing approx 400 AD:

"What then is time? If no one asks me, I know what it is. If I wish to explain it to him who asks, I do not know."

"How can the past and future be, when the past no longer is, and the future is not yet? As for the present, if it were always present and never moved on to become the past, it would not be time, but eternity."

Haha, Augustine warned us to be wary of mathematicians: :-)
...he used to warn that one should avoid astrologers who combine science and horoscopes. (Augustine's term "mathematici", meaning "astrologers", is sometimes mistranslated as "mathematicians".) According to Augustine, they were not genuine students of Hipparchus or Eratosthenes but "common swindlers".

Augustine also said: "People travel to wonder at the height of mountains, at the huge waves of the sea, at the long courses of rivers, at the vast compass of the ocean, at the circular motion of the stars; [at the physics of the universe]; and they pass by themselves without wondering."

mmm
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Re: Apparent Paradoxes

Unread postby sketch1946 » Mon Mar 27, 2017 10:02 pm

It's easy when using a tape measure or other rule, to think how you could measure more and more accurately if you had a magnifier.

But if you get bigger and bigger magnifications, some odd things start to happen, you start to see that the apparently finest lines on the rule are actually quite rough, at higher magnifications you can see only say the left edge of the zero mark, is this zero, or is zero in the middle of the wide black splodgy mark?

At higher magnification still you would see a quite irregular edge, so you would have to guess an average maybe of the most left parts and the most right parts **of the edge of the zero mark...so reality is a lot messier when you start to zoom in...

Imagine you had a scanning electron microscope, you might be able to identify the molecules that make up this hypothetical 'edge'...

A thought experiment then, going down to molecular size, where could this edge be? if it was say at the middle of a particular molecule, it would be impossible to make a cut here, molecules would have to shift either to left or right, but like looking at the sun, where is the edge?

Some sort of average must be applied, so there is no such thing as 'exactness' in the mathematical sense in the physical world.

If you tried to define the left edge of our rule as the centre of a molecule, how would you deal with the fact you can't measure it in any way, since it vibrates.. ie it moves... you could say the left edge is the furthest left that an electron shell could be found at, but a passing charge might attract the electrons away from the theoretical mark.

A thought exercise, imagine an universe with only two molecules. 'outside' them is nothing. Emptiness. how could you measure the size of this universe? Same problem as above, you could take an average of the size of one molecule and guess that the molecules were separated by x number of 'electron' widths, but what is one electron width exactly? Same problem.

If you gave one electron a push so that its kinetic energy was slightly higher than the gravitational force of the other atom, then this energised molecule would move away for ever, it would never be attracted back to the other molecule.

If you got a heap of molecules and gave them all a push in a radial direction, then they would never, ever come back, or ever touch each other...no stars, no galaxies, .... so much for the Big Bang.

So maths has things like a straight line, it has , concepts, points, infinity, progressions that unless they have an application to reality, are really in a virtual world that has nothing much to do with the world we live in...

Have you ever pondered what is exactly up... in a building a few metres wide, the left and right walls might be parallel, but what if the build was a half the width of the earth, would you make the walls parallel? What does a straight line mean? what does a line mean? What is half of a molecule? Is half of a strip of DNA, or half of a haemoglobin molecule useful for anything.... Look out at nature if you can see it near you, at the sea... what is the level of the top of the waves? where is the edge of the beach, what is the diameter of the sun, from which direction is the wind? How can you accurately measure the distance from the sun to a planet when it's always changing, and being affected by everything else in the solar system? Can it be inferred that tomorrow will be the same as today?

How can we be sure our model of the universe is valid? it could be infinite, or expanding, or contracting, or some other geometry, it all depends on how well we understand the sun, and the physics of red-shifting... are all suns the same? How well do we understand intergalactic space? How does light move through intergalactic plasma without being affected in some way? Do all rules of physics apply across the universe? How can we be sure? What is half way to something if space itself is expanding? Where does the solar 'wind' go to? Where do all the solar 'winds' of all the 100 billion stars of the galaxy go to? What about all the solar winds of all the stars in the universe? Where do all the charged particles of the intergalactic cosmic 'rays' come from? Why are they so energetic... where does this energy come from... etc....
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Re: Apparent Paradoxes

Unread postby webolife » Tue Mar 28, 2017 7:37 am

Back to Keith Ness,
The reason I find the finite universe more simply explained is this:
All objects or sets of objects are definable by the realm they occupy, beyond which they are "not there".
So it may be that "space" [or nothingness perhaps] beyond the universe is conceptually infinite, but the universe is a set of objects thus defined. There is no doubt circularity in all such reasoning, so it comes down to premises, not logic, to be the "pillars" of scientific thought. The universal "field", understood in this framework, can thereby act upon it's contained parts/objects in an discretely measurable way, which is the basis for all observation. Furthermore, fields within the universal field [zooming down to the subatomic level] can thus be conceptualized to operate according to the same "laws" as the [astronomic level] universe.
I'm curious about your using a "time before x" argument instead of spacial. How is "time" defined? {Rhetorical}
Truth extends beyond the border of self-limiting science. Free discourse among opposing viewpoints draws the open-minded away from the darkness of inevitable bias and nearer to the light of universal reality.
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Re: Apparent Paradoxes

Unread postby sketch1946 » Wed Mar 29, 2017 8:59 am

When it comes down to knowledge, and speculations about things in physics, what's wrong with "I don't know"?

In databases, there are large volumes of philosophy written about data integrity, about how each field must be designed to hold information with an appropriate amount of precision.. but some things are just fundamentally unknowable.. it's a subtle thing, but an information storage system with accurate information stored in it is really just an impossible goal, a hopeful dream unrelated to practical reality, so information systems, however speedy and massive and complex, are still just a **copy of reality... even the various Governments find it hard to pin people down with an ID, and to keep the other fields updated and accurate....Database gurus hate to have an empty field, to say "I don't know"....

Storing numbers in computers, and doing calculations without systematic errors is almost impossible:
How many digits are needed to store the number of atoms in the universe? Scientists can estimate a number and store it into a database, whatever number is stored, it is still a copy of the real number, in the database storing the best guess is still an estimation, in reality the number of atoms is ***unknowable, even if objectively it might seem to be some number 'n'....It's the same with any other information, there is the number in the database, and then the real number...

Eddington 'believed' a big number, and stated his 'belief' in a value for the number of protons in the universe:
"In the 1938 Tarner Lecture at Trinity College, Cambridge, Eddington averred that:

"I believe there are 15 747 724 136 275 002 577 605 653 961 181 555 468 044 717 914 527 116 709 366 231 425 076 185 631 031 296 protons in the universe and the same number of electrons."

Of course nowadays with computers we can be a lot more precise than that....

The problem with mathematics is the word 'real' applied to a 'real number' has a special meaning, in contrast to 'irrational numbers'... mathematics is constantly creating exotic new mathematical entities, new logic, it even has multiple infinities, hyper-real numbers, infinitesimals, surreal numbers, etc:
Numbers have birthdays, and a super big number is called 'No' :-)

Whenever something is discovered in maths that leads to paradoxes, the 'rules' of mathematics can be adapted to fix the problem:
Somewhere in the years after Eddington, mathematics as a tool of physics seems to have left the planet... drifted off into a sort of metaphysical ecstasy... progressively proving more and more esoteric and mysterious and telling totally impractical and improbable stories like kids in the playground saying "my daddy is bigger than your daddy...", this sort of intense metaphysical indulgence can be enough to drive brilliant mathematicians to a sort of frenzy of surreal creativity:

"Lamenting how their amazing discoveries drove them crazy, <...> George [Cantor] went near insane grappling with the concept of infinity, and Kurt [Gödel]'s mental stability deteriorated as he grappled with how George's problem lead to non-provable truths."

"Although not really a bad introduction to the work of these two fellows <...> First, there was a misplaced cause-and-effect message. Yes, both men had mental problems which ultimately became true mental illness. But it's highly questionable that their work was the primary cause. We also come away from the program hearing that Kurt and George had made discoveries of near mystic profundity shaking the very foundations of truth and the mind of man."

"<...>The key point is that for any integer, n, you have both a counting number - which is n - and its double, 2n, which is a unique even number. So there is a one-to-one correspondence between counting numbers and the even numbers, and half of a countable infinity is still a countable infinity!"

"Another weird property is that if you combine a countably infinite set to another countably infinite set, the new set is also countably infinite. Take the natural numbers (ergo, 1, 2, 3, etc.). Group them with zero and the negative numbers (-1, -2, -3, ...) and what you have is the set of integers. But you can prove the number of integers is the same as the number of counting numbers simply by setting up the following one-to-one correspondence. And although it's a little more complex, George showed that the fractions - rational numbers - are also countably infinite."

"<....>What stamped George as a great mathematician is he realized he didn't have to worry about the strange results (although some of his colleagues did). He just accepted them, and with them he invented the concept of cardinality. Essentially the cardinality is simply the size of a set.

For finite sets determining cardinality is simple. That's just the number of elements in the set. For infinite sets, though, it's a bit more abstract. Two sets (and this includes finite sets) have the same cardinality as long as their elements can be paired off in a one-to-one correspondence. Therefore all countably infinite sets - the even numbers, odd numbers, integers, fractions or whatever - also share the same cardinality. Since infinity is not a number, George designated the cardinality of a countably infinite set by the a symbol called aleph null, ℵ0

"In mathematics, "infinity" is often treated as a number (i.e., it counts or measures things: "an infinite number of terms") but it is not the same sort of number as natural or real numbers. Georg Cantor formalized many ideas related to infinity and infinite sets during the late 19th and early 20th centuries."

"The system of hyperreal numbers is a way of treating infinite and infinitesimal quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers greater than anything of the form

1 + 1 + ⋯ + 1 (for any finite number of terms).

Such numbers are infinite, and their reciprocals are infinitesimals."

"This shows that it is not possible to use a generic symbol such as ∞ for all the infinite quantities in the hyperreal system; infinite quantities differ in magnitude from other infinite quantities, and infinitesimals from other infinitesimals."

"Similarly, the casual use of 1/0 = ∞ is invalid, since the transfer principle applies to the statement that division by zero is undefined. The rigorous counterpart of such a calculation would be that if ε is a non-zero infinitesimal, then 1/ε is infinite."

"How many angels can dance on the head of a pin?" has been a major theological question since the Middle Ages.

According to Thomas Aquinas, it is impossible for two distinct causes to each be the immediate cause of one and the same thing. An angel is a good example of such a cause. Thus two angels cannot occupy the same space. This can be seen as an early statement of the Pauli exclusion principle. (The Pauli exclusion principle is a pillar of modern physics. It was first stated in the twentieth century, by Pauli.)

However, this does not place any upper bound on the density of angels in a small area, because the size r of angels remains undefined and could possibly be arbitrarily small. There have also been theological criticisms of any assumption of angels as complete causes..."
http://www.improbable.com/airchives/paperair/volume7/v7i3/angels-7-3.htm

Back to the problems of how big is infinity? Or how small is one divided by infinity?
"Concerns about the soundness of arguments involving infinitesimals date back to ancient Greek mathematics, with Archimedes replacing such proofs with ones using other techniques such as the method of exhaustion. In the 1960s, Abraham Robinson proved that the hyperreals were logically consistent if and only if the reals were. This put to rest the fear that any proof involving infinitesimals might be unsound, provided that they were manipulated according to the logical rules that Robinson delineated."

The simplest and most boring "proof" in the history of mankind began with Aristotle. But it shows the difference between a proof and a theorem.
All men are mortal.
Socrates is a man.
Socrates is mortal.

In this case the theorem is the final sentence. "Socrates is mortal." That's what you have proven. The proof is the list of sentences that lead to the final statement.

In essence then a proof is a list of statements arrived at by a given set of rules. Whether the theorem is in English or another "natural" language or is written symbolically doesn't matter. What's important is a proof has a finite number of steps and so uses finite number of symbols.

Now it is true that the number of symbols is open ended. If you don't want to use English you can use mathematical and logical symbols. In that case, you can have an unlimited number of variables which can be written as x0, x1, x2, ... and so on forever. But at most we can only use a countably infinite number of symbols. So any theorem must be shorter than a countably infinite number of steps and the limit of symbols is likewise countably infinite.

So no matter how many theorems we prove, books we write, or politicians give speeches, the symbols used will always make at most a countably infinite list. So at most there are only a countably infinite number of mathematical theorems and so only a countably infinite set of proofs.

OK. Let's take the set of natural numbers and make the power set. That is make all possible subsets of the natural numbers.

Then pick a number, any number. We'll pick the number "1". Then for every subset it will be either true or false that "1" is a member of that particular set. So there is at least one true mathematical statement associated with each subset of the natural numbers.

But wait. How many truths do we have? Well, since the number of subsets of the natural numbers is uncountably infinite, there must be an uncountable number of true mathematical statements.

But remember what we said above? We only have a countable number of theorems. So there are not enough theorems to prove all the true mathematical statements.

<...>[so] there are true mathematical statements that can't be proven.

http://www.coopertoons.com/education/di ... ument.html

Having wrestled with all these imponderables, how to manipulate them in a computer?
A subset of possible information about a person or event can be entered correctly into a database, and then that person can inconveniently die... then, as soon as possible, the now inaccurate information has to be 'updated'..... in another instance, information about an event can be entered into the database, but might be entered inaccurately, there is then a mismatch between reality and the stored information... this inaccurate information has to be corrected to keep a good match of this subset of reality stored in the database...

In storing data, there are volumes written on the special meaning of 'I don't know'.... special esoteric deep significance is attached to the value 'NULL'... or 'I don't have a clue'.. such that a computer querying the database for information is not allowed to assume that one NULL is equal to another NULL, since the values are unknown, therefore logically the equality of one unknown with another unknown can't be valid...

"Now, the caveat with SQL's NULL is that it doesn't behave like an absent value. It is the UNKNOWN value as others have also explained. This subtle difference has severe impact on a variety of operations and predicates, which do not behave very intuitively if you're not aware of this distinction...."

https://www.quora.com/What-is-the-signi ... ULL-in-SQL

Then there's the problem of fitting infinites into finite machines:
How can a number stored in binary base 2 be compared to a number stored in base 10? There are endless problems in these systems trying to get around the fact that collecting information is like collecting stamps... fascinating, illogical and addictive:
"
[In a computer, generally]
"The reason for the imprecision is the nature of number bases. In base 10, you can't exactly represent 1/3. It becomes 0.333... However, in base 3, 1/3 is exactly represented by 0.1 and 1/2 is an infinitely repeating decimal (tresimal?). The values that can be finitely represented depend on the number of unique prime factors of the base, so base 30 [2 * 3 * 5] can represent more fractions than base 2 or base 10. Even more for base 210 [2 * 3 * 5 * 7].

This is a separate issue from the "floating point error". The inaccuracy there is because a few billion values are spread across a much greater range. So if you have 23 bits for the significand, you can only represent about 8.3 million distinct values. Then an 8-bit exponent provides 256 options for distributing those values. This scheme allows the most precise decimals to occur near 0, so you can almost represent 0.1.

"....with an infinite number of decimals, I can for example express 'one third' exactly. The real-world problem is that not physically possible to have "an infinite number" of decimals or of bits."

http://stackoverflow.com/questions/1089 ... -in-binary

Built into many databases however are often many incorrect assumptions, yet a computer can still crank the numbers with mindboggling speed... common operations are tweaking different parameters to make a believable Big Bang, or other incredible feats of computation, but it still remains that these numbers are just numbers but not reality...

Mathematical ***systems are tools that are appropriate for analysis, for inferring new numbers from other numbers, creating new information, all of which must be meticulously stored with the appropriate precision, and continually checked to make sure each number or statement still matches reality, apparent paradoxes are example of when mathematics, language or logic ***fails.

An example of a famous paradox:
"Epimenides the Cretan says, 'that all the Cretans are liars,' but Epimenides is himself a Cretan; therefore he is himself a liar. But if he be a liar, what he says is untrue, and consequently the Cretans are veracious; but Epimenides is a Cretan, and therefore what he says is true; saying the Cretans are liars, Epimenides is himself a liar, and what he says is untrue. Thus we may go on alternately proving that Epimenides and the Cretans are truthful ***and untruthful."

The Epimenides paradox is the same principle as psychologists and sceptics using arguments from psychology claiming humans to be unreliable. The paradox comes from the fact that the psychologists and sceptics are human themselves, meaning that they state themselves to be unreliable. This means that psychology is not a science.

If light indeed travels at a constant speed, and if it has done so since the beginning, then nobody can get around the fact that the observed universe is out of date!

If something happened to our nearest neighbour star, we wouldn't know about it for four years... say for instance if our galaxy blew up ***today, in an uncontrolled chain reaction, a sort of supernova atom bomb.. and extinction was heading our way from the centre of the galaxy outwards.. we would still be sitting here smugly theorising and calculating and measuring and sending probes to the nearest likely planet, we would be thinking we were fine, we'd be thinking we're OK for 50,000 years before the big crunch arrived....

So everything out there is out of date... old data...

...or maybe the current model is all wrong, and we need a new model, ie new physics, or non-constant constants: :-)

"...the future world model will reveal [a] deep connection between fundamental physics and cosmology:
”There may even be some big surprises: time variation of the constants or a new theory of gravity that eliminates the need for dark matter and dark energy”

We need to sort out some creative answers to some current paradoxes:
"•Gravitation field energy paradox: in the framework of the Einstein’s geometrical gravity theory (General Relativity) there is no physical concept of the energy-momentum density of the gravitational field (also there is no physical concept of the energy quanta of the gravitational field), though field energy exists for all other fundamental physical interactions.
•1st Harrison’s paradox: physics of space expansion contains such puzzling phenomena as continuous creation of vacuum, violation of energy conservation, violation of limiting velocity by receding galaxies.
•2nd Harrison’s paradox: the cosmological redshift in expanding space is not the Doppler effect, but it is a new physical phenomenon which [has not been] tested in the lab, the global gravitational cosmological redshift should be taken into account.
•Hubble-de Vaucouleurs’ paradox: in the expanding space the linear Hubble law is the fundamental consequence of the homogeneity, however modern observations reveal existence of strongly inhomogeneous fractal large-scale galaxy distribution at scales at least up to 100 Mpc, while the linear Hubble law starts from 1 Mpc, i.e. just inside inhomogeneous structure.
Paradoxes of cosmological physics in the beginning of the 21-st century
Yurij Baryshev
https://arxiv.org/pdf/1501.01919.pdf
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