Another major scientific breakthrough!

[HoreTore]

2012 shapes up to be a good year for science. First the Higgs-Boson was found, now we have a solution to the abc-conjucture (http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20I.pdf).

Do I understand any of it? Heck no! I do recognize some of its implications though, like giving solutions to certain Diophantic problems. Exciting nonetheless! I'm having a meeting with my old maths professor in October, I'll try to harass him for some answers... In the meantime, does anyone here understand some of it?

[SoFarSoGood]

Not 'science' as such but number theory. Nor is the Higgs - Boson 'found'. It is probable.

[Fragony]

Am I stupid because I really don't understand that or just stupid

[HoreTore]

Am I stupid because I really don't understand that or just stupid

This is the absolute top of mathematics Frags, I don't understand a word of it. Well, I can recognize the symbols and such, but I'm completely clueless as to what he's saying. Kinda like reading dutch: I understand the letters and can spot one or two words, but I have no chance to udnderstand what it means.

I read an article about it which was enlightening, but it was in Norwegian. I may be able to find one in english...

[Tellos Athenaios]

Not 'science' as such but number theory.

Very much science, then.

Anyway if the proof stacks up this would indeed signal party time.

[Major Robert Dump]

I thought this would be about the new monkey they found, the one who looks like Gene Wilder. I love Willy Wonka

[Tellos Athenaios]

Am I stupid because I really don't understand that or just stupid

Here is a gentle introduction to the subject:
http://www.ega-math.narod.ru/Liv/Goldfeld.htm

[Moros]

Also other major scientific news:
http://www.huffingtonpost.com/2012/09/15/single-molecule-images-ibm-chemical-bonds_n_1884818.html

IBM was able to photograph bounds within a molecule, continuing and improving on their older pictures of molecules.

[HoreTore]

Also other major scientific news:
http://www.huffingtonpost.com/2012/09/15/single-molecule-images-ibm-chemical-bonds_n_1884818.html

IBM was able to photograph bounds within a molecule, continuing and improving on their older pictures of molecules.

If this had been discovered by CERN, I'm sure a number of clueless politicians would've called for an end to such "useless research". It's a lot harder to call the research a waste of money when it's done by a commercial company though.

[Sarmatian]

If this had been discovered by CERN, I'm sure a number of clueless politicians would've called for an end to such "useless research". It's a lot harder to call the research a waste of money when it's done by a commercial company though.

Well, in all fairness, commercial companies aren't funded with taxpayer's money for the most part.

[HoreTore]

Well, in all fairness, commercial companies aren't funded with taxpayer's money for the most part.

That's completely besides the point. I think you may have to reread my post...

The point is that this kind of research generates profit. If it didn't, IBM wouldn't be doing it. In other words, not funding this kind of research is a waste of tax payer money, while funding it generates more money.

[Moros]

Well, in all fairness, commercial companies aren't funded with taxpayer's money for the most part.

Where does the money come from? The same people. Who aren't always happy with what they get for their money either. Do you really think goverments are the only organisations that rip us from our money? ~;)

[Papewaio]

Mathematics isn't science.

It's like confusing weights with weight lifting.

[Beskar]

There was the discovery of a new species of human (http://www.bbc.co.uk/news/science-environment-19184370) a while back.

It looks like even old text books can be wrong.

[Philippus Flavius Homovallumus]

Mathematics isn't science.

It's like confusing weights with weight lifting.

Mathematics is the only Science.

OK, so what I want to know is did this chap do it properly using chalk and tables, or did he cheat and use a computer?

[Moros]

Mathematics isn't science.

It's like confusing weights with weight lifting.
I guess it depends on what you'd call science. Just out of interest what do and don't you consider science and why?

[Papewaio]

Science deals with the study of nature, the world around us. Mathematics also includes the imaginary.

You also can do maths as an arts degree...

[HopAlongBunny]

Mathematics isn't science...

But much to the chagrin of mathematicians, scientists seems to always find a "real world" use for it:p The search for a theorem that is not in any way useful continues...

Nature article on the original topic:

http://www.nature.com/news/proof-claimed-for-deep-connection-between-primes-1.11378

Link to Mochizuki's blog:

http://www.kurims.kyoto-u.ac.jp/~motizuki/papers-english.html

[Fragony]

Science deals with the study of nature, the world around us. Mathematics also includes the imaginary.

You also can do maths as an arts degree...

Got to agree here, from what I read it just means the tools we use are off, if you can prove you can make two cubes out of one mathemetically there is something wrong with math itself.

[Philippus Flavius Homovallumus]

Science deals with the study of nature, the world around us. Mathematics also includes the imaginary.

You also can do maths as an arts degree...

No - just no.

Science is a method, not a discipline.

Mathematics conforms to the method, therefore it is a Science.

[HoreTore]

Science deals with the study of nature, the world around us.

That would be natural science, ie. biology, physics and chemistry, not science.

[Philippus Flavius Homovallumus]

I feel utterly vindicated.

[rvg]

No - just no.

Science is a method, not a discipline.

Mathematics conforms to the method, therefore it is a Science.

This is false. Mathematics most definitely does not operate on Scientific Method. Mathematics lacks the necessary empirical component.

[Tellos Athenaios]

This is false. Mathematics most definitely does not operate on Scientific Method. Mathematics lacks the necessary empirical component.

I don't think you understand what "science" or "empirical" mean.

[rvg]

I don't think you understand what "science" or "empirical" mean.
I don't think you understand what "science" or "empirical" mean.

[Tellos Athenaios]

Let's start with the easy one, shall we? Empirical: can be observed, or shown by way of experiment.

The fact that prime numbers cannot be factorised into smaller primes (which follows from the definition of prime) can be lend credibility using empirical evidence of taking a prime number, say, 13 and trying to find an integer factor larger than 1 and smaller than this number (13). It turns out this is impossible which yields an empirical data point backing this statement.

You are now invited to dispel the notion of empirical evidence backing the work of Mathematicians, ignoring the body of history in which Mathematics was founded on observed (empirical) curiosities. Please, it'll be fun.

[Centurion1]

as a dual math major im very excited by this news it is a big deal. my professors are all likely to be in highly aroused states for the remainder of the year.

[Philippus Flavius Homovallumus]

This is false. Mathematics most definitely does not operate on Scientific Method. Mathematics lacks the necessary empirical component.

This is what happens when you atomise the Academy and muck around with the definitions of words like "science".

I'm sympathetic to HoreTore's irritation, but even so it's nice not to be the one on the receiving end of this.

Theoretical Physics is a Science - or at least it was last time I checked, that's just a branch of Mathematics.

[rvg]

It turns out this is impossible which yields an empirical data point backing this statement.
It's not empirical data. It cannot be physically measured.

[Tellos Athenaios]

It's not empirical data. It cannot be physically measured.

So all I have to do is introduce "measurable" units that your mind feels more comfortable with? Thereby giving you a direct physical example of Math at work? Alright, the cm will do nicely: substitute 1cm for 1, therefore 13cm for 13. Using this convention, factorising 8 into its prime factors is simply a matter of taking a length of 8cm and dividing it into parts each of 2cm. So tell me now how can you not physically demonstrate that 13cm cannot possibly be split into smaller, equal parts which are a whole (integer) multiple of the cm and larger than 1cm? How is that not empirical?

You must have been asleep when people tried to explain units of measurement to you.

[rvg]

So all I have to do is introduce "measurable" units that your mind feels more comfortable with? Thereby giving you a direct physical example of Math at work? Alright, the cm will do nicely: substitute 1cm for 1, therefore 13cm for 13. Using this convention, factorising 8 into its prime factors is simply a matter of taking a length of 8cm and dividing it into parts each of 2cm. So tell me now how can you not physically demonstrate that 13cm cannot possibly be split into smaller, equal parts which are a whole (integer) multiple of the cm and larger than 1cm? How is that not empirical?

How is it empirical? Does a number exist in a physical world?

[Tellos Athenaios]

How is it empirical? Does a number exist in a physical world?

A number is a quantity, nothing more.

[Fragony]

A number is a quantity, nothing more.

A quantity is a quantity, seems like numbers just don't cut it

[Tellos Athenaios]

A quantity is a quantity, seems like numbers just don't cut it

How?

[Fragony]

How?

From your link, a cube can apparentlty by devided into two cubes, that is not possible yet can be proven mathematically

[Tellos Athenaios]

From your link, a cube can apparentlty by devided into two cubes, that is not possible yet can be proven mathematically

A cube cannot be partitioned in this way, that is exactly what Fermat's Last Theorem states (when considering cubes n = 3, therefore n > 2) and it is mathematically proven to be impossible. I think you misread?

[Fragony]

A cube cannot be partitioned in this way, that is exactly what Fermat's Last Theorem states (when considering cubes n = 3, therefore n > 2) and it is mathematically proven to be impossible. I think you misread?

Possibly, but I think you misunderstood, read again

[Tellos Athenaios]

Possibly, but I think you misunderstood, read again

Which bit? From "my link":

It is impossible to separate a cube into two cubes, or a biquadrate into two biquadrates, or in general any power higher than the second into two powers of the like degree; I have discovered a truly remarkable proof which this margin is too small to contain.

For the "biquadrate" (fourth power) case, Fermat's earlier assertion is sufficient to imply the later one: if two fourth powers cannot sum to a perfect square, they cannot sum to a fourth power either (since any fourth power, say w4, is also a perfect square, namely, the square whose side measures w2). But Fermat was asserting much more. In modern notation Fermat's assertion – known to mathematicians as Fermat's last theorem, or FLT for short – states that the equation xn + yn = zn has no solution if x, y and z all are positive integers and n is a whole number greater than 2.

[Fragony]

Which bit? From "my link":

That's too hard for me, you are talking to a mavo c here. But it's pretty obvious that it are the actual methods that are on trial here. If you can be a wizard with the numbers the numbers are wrong, pure and simple

[rvg]

A number is a quantity, nothing more.

Where in nature can I find 731?

[Montmorency]

I see it on my screen right here.

[rvg]

I see it on my screen right here.
Can you weigh it?

[Montmorency]

Some multiple of 9.10938188 × 10^-31 kilograms, presumably.

[rvg]

Some multiple of 9.10938188 × 10^-31 kilograms, presumably.
And that's the weight of what?

[Montmorency]

Lies.

[rvg]

Lies.
Indeed.

[Montmorency]

Electrons, my mistake.

Lies are heavier.

[Askthepizzaguy]

They make baby jesus cry.

Weigh baby Jesus for me.

Also, God.

[HoreTore]

Where in nature can I find 731?

This is a fundamental misconception of what mathemathics is. An extremely common one, but still fundamentally wrong.

Mathematics is not abstract. Not in any way whatsoever. One can use abstract thinking to solve problems, like in every other scientific field, but the basis of mathematics will always be the natural world. Numbers exist because they are the names we have given to sizes we find in the world.

731 is easily found, just buy 731 apples, put them on the table before you and voila.

[rvg]

This is a fundamental misconception of what mathemathics is. An extremely common one, but still fundamentally wrong.

Mathematics is not abstract. Not in any way whatsoever. One can use abstract thinking to solve problems, like in every other scientific field, but the basis of mathematics will always be the natural world. Numbers exist because they are the names we have given to sizes we find in the world.

731 is easily found, just buy 731 apples, put them on the table before you and voila.

No, no. I'm not looking for 731 apples, or 731 electrons or 731 widgets.
I'm looking for 731.

[HoreTore]

No, no. I'm not looking for 731 apples, or 731 electrons or 731 widgets.
I'm looking for 731.

Then you are looking for something different than what mathematicians look for.

It's like saying astronomy is unscientific because you can't see any stars in your waste basket.

[rvg]

Then you are looking for something different than what mathematicians look for.

I'm sorry, I didn't realize that all this time mathematicians had apples in mind. Thank you for enlightening me.

[HoreTore]

I'm sorry, I didn't realize that all this time mathematicians had apples in mind. Thank you for enlightening me.

No problem!

[Populus Romanus]

Can you weigh it?Weight is irrelevant. The photon, for instance, has no mass at all. Still exists.

[rvg]

Weight is irrelevant. The photon, for instance, has no mass at all. Still exists.

Love also exists, but it's not an empirical object. Anyway, the idea of numbers being anything but abstract is so stupid, I'm not even gonna discuss it. Next thing I'll be hearing is that the Earth is flat.

[Philippus Flavius Homovallumus]

Love also exists, but it's not an empirical object. Anyway, the idea of numbers being anything but abstract is so stupid, I'm not even gonna discuss it. Next thing I'll be hearing is that the Earth is flat.

numbers aren't abstract, they're an abstraction - they are a sign used to represent a concrete reality.

[HoreTore]

Love also exists, but it's not an empirical object. Anyway, the idea of numbers being anything but abstract is so stupid, I'm not even gonna discuss it. Next thing I'll be hearing is that the Earth is flat.

No need for me to say much more, I'll hand the microphone over to PVC:


numbers aren't abstract, they're an abstraction - they are a sign used to represent a concrete reality.

Spot on.

Numbers are a generalization of a concrete reality.

[Papewaio]

That would be natural science, ie. biology, physics and chemistry, not science.

Everything else is stamp collecting...

[Philippus Flavius Homovallumus]

Everything else is stamp collecting...

Ugh.

At least you're being an equal-opportunities knucklehead.

Lets go through the hierarchy of Philosophy again: we progress downwards, each branch being a descendant of higher branches. So, at the top we have metaphysics, of which the highest part is epistemology - once we have decided that we accept that we can now things, even though we can't conclusively prove them guess what comes next?

Mathematics - without the basic principles of which we could not operate logic.

[Papewaio]

Paraphrase was from Rutherford and I normally sig when I'm whimsical.

Philosophy isnt science neither is mathematics.

Mathematics is a subset of philosophy.

Science a subset of mathematics.

However a container is not the fluid and not all rectangles are squares.

If a superset = a set 1:1 then they are the same. Maths is science plus something more.

[Philippus Flavius Homovallumus]

Science is just a method though - that's why we call them the "Natural Sciences" - Mathematics is most certainly a Science though, because it uses the Method.

More to the point, Theoretical Physics is just Mathematics, so if Mathematics isn't a Science then neither is Theoretical Physics. If Theoretical Physics isn't a Science then you're about 30 seconds from our entire conception of knowledge collapsing.

I'm sorry, it doesn't wash.

[Papewaio]

The difference with science is that your set of proofs are circumsribed by reality. You can't prove something is a valid theory without testing it against something tangible. For all the grief experimental scientists get from theoretical the thing that separates science from philosophy is that proofs need to be tested not just talked about.

Mathematics has the ability to have sets of proofs not constrained to the current universe we inhabit. It is possible to have mathematical models with no 'real' world analogue to test against.

Maths has the ability to deal with things outside that which is physical. Drop a negative apple on my head and I will place maths lower on the totem pole besides science.

[Philippus Flavius Homovallumus]

The difference with science is that your set of proofs are circumsribed by reality. You can't prove something is a valid theory without testing it against something tangible. For all the grief experimental scientists get from theoretical the thing that separates science from philosophy is that proofs need to be tested not just talked about.

Mathematics has the ability to have sets of proofs not constrained to the current universe we inhabit. It is possible to have mathematical models with no 'real' world analogue to test against.

Maths has the ability to deal with things outside that which is physical. Drop a negative apple on my head and I will place maths lower on the totem pole besides science.

This doesn't make Mathematics "not Science".

[Fragony]

This doesn't make Mathematics "not Science".

Sure it does, it's just a tool. Refining the tool is something different than science, it's just what's at hand at the time. That mathematicians are incredibly smart doesn't change that they are making tools.

[HoreTore]

Mathematics has the ability to have sets of proofs not constrained to the current universe we inhabit.

Mathematics is derived from the "current universe we inhabit", and since it doesn't add something else, it can't magically start dealing with things outside it.

Otherwise, I'm going to need some proof to back that up. The proof in the OP, for example, very much deals with "the current universe".

[Papewaio]

Write out a Googolplex all it's zeros without using scientific notation, cut out a square of area i, pi... there isn't enough atoms in the universe to create a circle that has a circumference exactly equal to pi.

[Fragony]

Write out a Googolplex all it's zeros without using scientific notation, cut out a square of area i, pi... there isn't enough atoms in the universe to create a circle that has a circumference exactly equal to pi.

That does not make sense at all, why would atoms in any way be related to pi

[Tellos Athenaios]

Write out a Googolplex all it's zeros without using scientific notation
So? A line has infinitely many points on it. Assign each 0 of the googolplex to a point on your line, and you can write out the number of zero's by drawing a line of length lim x->infinity = 10^(10^100)/x = 0. Infinity is a beautiful concept, especially once you realise some kinds of infinity are more infinite than others! Then you realise what you've actually done by the above definition is draw an uncountably infinite number of zeros, which is far more than was ever asked.


cut out a square of area i

You are conflating scalars and vectors (pairs). You ask to cut out a square of "left" or "right".


pi... there isn't enough atoms in the universe to create a circle that has a circumference exactly equal to pi.

There is more than enough, provided you start qualifying your dimensions first. Your problem has nothing to do with Math, everything to do with the fact that we are not able to manufacture anything to the degree of precision (infinite) which "exactly" requires. This however does not invalidate pi: in fact pi is very physically manifest in lots of ways.

[Gregoshi]

That does not make sense at all, why would atoms in any way be related to pi
Ask Carl and his apple pi:


https://www.youtube.com/watch?v=gh4F5BQ8hgw&feature=related

[Papewaio]

So? A line has infinitely many points on it. Assign each 0 of the googolplex to a point on your line, and you can write out the number of zero's by drawing a line of length lim x->infinity = 10^(10^100)/x = 0. Infinity is a beautiful concept, especially once you realise some kinds of infinity are more infinite than others! Then you realise what you've actually done by the above definition is draw an uncountably infinite number of zeros, which is far more than was ever asked.

Now write it out and you will run out of atoms in this universe before finishing it. It is a mathematical concept bigger then this universe.

"You are conflating scalars and vectors (pairs). You ask to cut out a square of "left" or "right"."

Again create a physical object for an imaginary number.

"There is more than enough, provided you start qualifying your dimensions first. Your problem has nothing to do with Math, everything to do with the fact that we are not able to manufacture anything to the degree of precision (infinite) which "exactly" requires. This however does not invalidate pi: in fact pi is very physically manifest in lots of ways."

Pi goes to infinity. Number of atoms in the universe are finite. Therefore it is not possible in this universe to create a physical 1:1 accurate disc that has exactly the ratio of 2pi.r atoms in the circumference to atoms in the radius. In fact the circle really isn't a circle it's a many sided polygon which is the best approximation a finite universe can make. Much better then a triangle made of 3 atoms but just as short from infinity.

Maths isn't restricted to the physical universe as science is. Maths can be purely theoretical, science requires physical experimentation. If we stepped between one universe and another, the atoms could have different sub atomic particles, forces, gravity constant etc The science outcomes would be different as the fundamental forces would be different. Maths on the other hand wouldn't need such a frame shift.