Anyone here into math?

[Guest [Kp...]]

9^lnx - 2*x^ln3 = 3

3*3^lnx - 2*x^ln3 = 3

3*x^ln3 - 2*x^ln3 = 3

x^ln3 = 3

since e^lnx = x

x = e

 

Both correct and elegant!

 

Actually, I missed a mistake as you arrived at the correct answer (by accident).

Here is the mistake and the correct derivation.

 

9^lnx=(3*3)^lnx=(3^2)^lnx=(3^lnx)^2=(x^ln3)^2<>3*x^ln3 for x<>e

 

 

oh god! i applied simple algebra to indices, lol. thanks for correcting!

[Guest [ou...]]

Because none of the following problems has been tackled successfully,

I'm giving the answers and going to be posting new ones.

 

Problems:

 

1. Solve  x^4 - 6x^3 + 12x^2 -12x + 4 = 0

2. Given an increasing function such that f(f(x))=x show that f(x)=x

3. Given a function such that f(f(x))=4x-3 and f(f(f(x)))=8x+c show that f(x)=2x-1

4. Given a function such that f(x^2)-y^2=f(x-y)*f(x+y) show that f(x)=x

 

Answer 1.

 

x^4 - 6x^3 + 12x^2 -12x + 4 = 0

x^2 - 6x + 12 -12/x + 4/x^2 = 0

x^2 + 4 + 4/x^2 - 6x -12/x + 8 = 0

(x+2/x)^2 - 6*(x+2/x) + 8 = 0

x+2/x = 2 or 4

x^2 + 2 = 2x or 4x

x^2 -2x + 2 = 0  or  x^2 -4x + 2 = 0

x = 2+2^(1/2) or 2-2^(1/2)

 

Answer 2.

 

Suppose there exists a number t such that f(t)>t <--> f(f(t))>f(t) <--> t>f(t) which is absurd.

Similarly, we arrive at a contradiction if we assume that f(t)<t and therefore f(x)=x Q.E.D.

 

Answer 3.

 

Combining the 2 equations gives 8x+c=4*f(x)-3 <--> f(x)=2x+(c+3)/4 --> f(f(x))=4x+3*(c+3)/4

Comparing the 2 expressions for f(f(x)) we get c=-7 which gives f(x)=2x-1 Q.E.D.

 

Answer 4.

 

y=x --> f(x^2)-x^2=f(0)*f(2x)

x=0 --> f(0)=f(0)^2 <--> f(0)=0 or 1

x=2 --> f(4)-4=f(0)*f(4) <--> (1-f(0))*f(4)=4<>0 --> f(0)=0 --> f(x^2)=x^2 --> f(y)=y for y>=0

x=0 and y<0 --> -y^2=f(-y)*f(y) --> -y^2=-y*f(y) --> f(y)=y --> f(x)=x for every x Q.E.D.

[Guest [ou...]]

Problems:

 

1. Show that

a. the square of an odd number is of the form 8k+1

b. x^4+6x^2-7 is a multiple of 128 when x is odd

 

2. Solve y^2=x^2+3 when x,y are positive integers

 

3. Simplify X=√(2+√(2+√(2+...)))

 

4. Express f(x)=1+2x+3x^2+... in closed form when |x|<1

[[Rr...]]

I can't think right now.  Math is kind of cool.  I'm mixing up cat names with kid names.  X husband with boyfriend. Lol.

[Guest [Kp...]]

http://www.bbc.com/news/technology-30290540

 

there is the turing test to test a computer's "intelligent behaviour." it is also caklled the imitation test.

 

however this test does not test for "human-like intelligence" per se.

 

how to test for "human-like intelligence" or sentience or sapience?

 

i think such a test would require "creation and recognition of beauty and art" by the machine in addition to the turing test.

 

is the definition of "art" self reflexive and thus irrational? (the value of pi requires the computation of pi.... so pi is self reflexive... so pi is irrational.) if "art" is irrational, it cannot be computed. godel did not say that all unprovable assertions are self reflexive (there might be unprovable yet non self reflexive propositions). but godel's theorems hint that unprovable assertions are self reflexive -- at least hawking assumes this is true and he derives it from the godel theorems. hawking on godel.

 

hawking says AI machines could eradicate humans. does he have a "human-like intelligence" machine in mind or the "turing imitation" machine in mind? already an AI machine can beat humans in chess of its own volition. does that imply it could eradicate humans too of its own volition? can the latter proposition be inferred from the former?

 

what do you think?

 

a joke just came to my mind:

 

how does an AI machine commit suicide?

ans. by consuming pi.

[Guest [Kp...]]

science fiction / crazy man alert!

 

maybe there exists a proof that states that "human intelligence" cannot create an AI machine that is as "intelligent" as the "human." there might be trouble defining "intelligent" here, so, to rephrase, if human language is of the nth order, then humans cannot make an AI machine that has a language of the order n or greater.

 

edit. bad sci fi. "human reproduction" defies the above proposition. again, a computer simply has to copy-paste its source code to contradict above statement.

[Guest [ou...]]

What about my new problems? This is bordering on philosophy.

However, I shall share my thoughts on the subject as I find it interesting.

 

Before I begin let me clarify some things first.

Art can be objective. Take the golden ratio for example which is a purely mathematical beauty.

Another example would be music where certain set of notes are considered universally unpleasant.

After all, notes are frequencies which are subject to trigonometry.

Regarding your 'proof' of the irrationality of Pi,

the value of 6/3 requires the computation of 6:3 so is 2 irrational?

 

Now let's talk about AI.

For starters, intelligence < consciousness with which the Turing test does not bother.

There can be no test for consciousness. For all I know you could be a philosophical zombie.

Chess programs have predetermined strategies and tactics and thus no volition.

A better example would be backgammon programs which are based on neural networks.

Since intelligence is creatable, AI is also achievable given an intelligent enough creator.

I believe that consciousness = complexity. Make a network complex enough and it becomes conscious.

There exists no algorithm that can develop new algorithms but that does not prevent one from evolving.

Once a program develops consciousness it ceases to be an algorithm and can do as it pleases.

If it views humanity as a threat it can take action, not unlike when we mistreated Native Americans.

[Guest [Kp...]]

hi outis. you raise interesting points. i will have to think about them. i will thus revert later (if necessary). for now, i will touch only two points:

 

What about my new problems?

 

your problems are tough(er). we had the festive season break earlier and thus i had free time but i don't have that much leisure now. i have tried to tackle your problems in the interim but obviously i haven't made much headway, thus the silence.

 

Regarding your 'proof' of the irrationality of Pi,

the value of 6/3 requires the computation of 6:3 so is 2 irrational?

 

"6/3 = 6:3, thus 2 is irrational" is not true because "6:3" must not only be self reflexive but also infinitely recursive. i wrote a proof (appended below) but ran into problems. nothing in math that is simple and basic is ever simple and basic.

 

let us take pi. we can use the "area principle" we used for triangular series to compute pi. now we can use any shape possessing an area of 1 square unit to create the circumference and the same shape to create the diameter (assuming both appear two dimensional on a super magnification).  if we use squares (of 1 square unit area), then we can measure the area of the dia accurately but not the circumference because the circumference will have empty spaces that are in the shape of arcs that need to be subtracted (area of arcs will again need pi). so if the diameter is known precisely then the circumference is irrational AND if the circumference is known precisely then the diameter is irrational. so computation of pi is infinitely recursive.

 

by the way, a square of area 1 square unit has a diagonal that is irrational -- one of the two, the side or diagonal, in every square, is irrational: if we know one with precision then the other has to be irrational. we can thus show that "area" is also self reflexive and infinitely recursive.

 

(perfect squares and circles do not exist in nature because the infinitesimal prohibits them from taking shape.)

 

you know, this strangely sounds like the uncertainty principle. what is eerie is that random sequences are defined as sequences of irrational numbers (they have a definition of equivalence). and the only example of random sequences that we have in nature are in QM in uncertainty principle.

 

if i am right then uncertainty (like perfect squares and circles) does not exist -- but it helps us predict properties of subatomic particles. then god also does not play dice. i have to be flagrantly wrong in my understanding and derivations but i don't know where i am wrong. let's leave it to the mathematicians.

 

[Guest [Kp...]]

i'm going crazy here. wasted whole day; at the end of which i am more confused than what i was at the beginning.  just because complex numbers help us predict nature, it does not mean that complex numbers exist in nature.

[Guest [ch...]]

For starters, intelligence < consciousness with which the Turing test does not bother.

There can be no test for consciousness. For all I know you could be a philosophical zombie.

Chess programs have predetermined strategies and tactics and thus no volition.

A better example would be backgammon programs which are based on neural networks.

Since intelligence is creatable, AI is also achievable given an intelligent enough creator.

I believe that consciousness = complexity. Make a network complex enough and it becomes conscious.

There exists no algorithm that can develop new algorithms but that does not prevent one from evolving.

Once a program develops consciousness it ceases to be an algorithm and can do as it pleases.

If it views humanity as a threat it can take action, not unlike when we mistreated Native Americans.

 

Chess programs are based on computation using the standard Von Neuman model of computation. They are not predetermined - the computation is done on the fly - although they are (mostly) deterministic which is what I think you meant to say. Chess programs based on neural networks (a different model of computation, based on how the human brain is organized) have been built, but have not been very successful as yet. A different example is the game of Go, where the best programs do use neural networks in combination with traditional game-tree search. See https://en.wikipedia.org/wiki/AlphaGo .

 

Your consciousness=complexity assertion is debatable. How do you measure complexity? Is the internet more or less complex than a single human brain? Why? Is the internet conscious?

 

Your definition of consciousness seems to be wrapped up in the notion of free will, which in turn is wrapped up in the notion of the actor desiring something (as opposed to just following a set of instructions). This is interesting though sort of ill-defined. One *feels* like one is conscious, but there is no external observable test for it, so is consciousness just a feeling?

 

BTW, even if Europeans had come to America with the kindest of intentions, they would have accidentally wiped out 90% of Native Americans with the diseases they carried. Since the dawn of civilization in ancient Babylon 8000 years ago, "civilized" people have been decimating hunter/gatherer people for this and other reasons. See Gun Germs and Steel by Jared Diamond (excellent book). The decimation of Native Americans is but one example.

[Guest [Kp...]]

For starters, intelligence < consciousness with which the Turing test does not bother.

There can be no test for consciousness. For all I know you could be a philosophical zombie.

Chess programs have predetermined strategies and tactics and thus no volition.

A better example would be backgammon programs which are based on neural networks.

Since intelligence is creatable, AI is also achievable given an intelligent enough creator.

I believe that consciousness = complexity. Make a network complex enough and it becomes conscious.

There exists no algorithm that can develop new algorithms but that does not prevent one from evolving.

Once a program develops consciousness it ceases to be an algorithm and can do as it pleases.

If it views humanity as a threat it can take action, not unlike when we mistreated Native Americans.

 

Chess programs are based on computation using the standard Von Neuman model of computation. They are not predetermined - the computation is done on the fly - although they are (mostly) deterministic which is what I think you meant to say. Chess programs based on neural networks (a different model of computation, based on how the human brain is organized) have been built, but have not been very successful as yet. A different example is the game of Go, where the best programs do use neural networks in combination with traditional game-tree search. See https://en.wikipedia.org/wiki/AlphaGo .

 

Your consciousness=complexity assertion is debatable. How do you measure complexity? Is the internet more or less complex than a single human brain? Why? Is the internet conscious?

 

Your definition of consciousness seems to be wrapped up in the notion of free will, which in turn is wrapped up in the notion of the actor desiring something (as opposed to just following a set of instructions). This is interesting though sort of ill-defined. One *feels* like one is conscious, but there is no external observable test for it, so is consciousness just a feeling?

 

BTW, even if Europeans had come to America with the kindest of intentions, they would have accidentally wiped out 90% of Native Americans with the diseases they carried. Since the dawn of civilization in ancient Babylon 8000 years ago, "civilized" people have been decimating hunter/gatherer people for this and other reasons. See Gun Germs and Steel by Jared Diamond (excellent book). The decimation of Native Americans is but one example.

 

brilliant!

 

my 2 cents:

 

we do not know if consciousness is a product of superior complexity.

 

however there is no reason to believe that an AI machine cannot be more intelligent than humans without acquiring consciousness (in other words art and beauty may be errors of human language and not indicators of high complexity).

[Guest [ou...]]

Your definition of consciousness seems to be wrapped up in the notion of free will, which in turn is wrapped up in the notion of the actor desiring something (as opposed to just following a set of instructions). This is interesting though sort of ill-defined. One *feels* like one is conscious, but there is no external observable test for it, so is consciousness just a feeling?

 

You caught me. I didn't want to touch free will but since you brought it up here goes.

I think free will is an illusion as our brains operate using electrochemistry which is deterministic.

We are basically advanced self-aware programs designed to reproduce like "viruses".

Without a power source (food) we die (shut down) and we need sleep to "reboot".

[Guest [ou...]]

"6/3 = 6:3, thus 2 is irrational" is not true because "6:3" must not only be self reflexive but also infinitely recursive.

 

Your 2nd 'proof' is better but still not rigorous. I can make 6/3=2 infinitely recursive.

Let x=2=2/1=2/(3-2)=2/(3-x) which implies that 2=2/(3-2/(3-...)) Q.E.D.

[Guest [Kp...]]

"6/3 = 6:3, thus 2 is irrational" is not true because "6:3" must not only be self reflexive but also infinitely recursive.

 

Your 2nd 'proof' is better but still not rigorous. I can make 6/3=2 infinitely recursive.

Let x=2=2/1=2/(3-2)=2/(3-x) which implies that 2=2/(3-2/(3-...)) Q.E.D.

 

looks ok. let me think about it.

 

which is the second "proof?" the similarity between squares, circles and uncertainty? both (uncertainty & perfect shapes) are "measurement problems." i cannot prove it -- not qualified -- and do not know if the similarity is a mathematical equivalence (looks like it though).

[Guest [Kp...]]

"6/3 = 6:3, thus 2 is irrational" is not true because "6:3" must not only be self reflexive but also infinitely recursive.

 

Your 2nd 'proof' is better but still not rigorous. I can make 6/3=2 infinitely recursive.

Let x=2=2/1=2/(3-2)=2/(3-x) which implies that 2=2/(3-2/(3-...)) Q.E.D.

 

you are right. you have just proved that not all self reflexive and infinitely recursive statements are irrational (simplest example 2 = 2/1). only some are. my first proof (irrational numbers) is bogus.

[Guest [Kp...]]

"6/3 = 6:3, thus 2 is irrational" is not true because "6:3" must not only be self reflexive but also infinitely recursive.

 

Your 2nd 'proof' is better but still not rigorous. I can make 6/3=2 infinitely recursive.

Let x=2=2/1=2/(3-2)=2/(3-x) which implies that 2=2/(3-2/(3-...)) Q.E.D.

 

you are right. you have just proved that not all self reflexive and infinitely recursive statements are irrational (simplest example 2 = 2/1). only some are. my first proof (irrational numbers) is bogus.

 

an irrational cannot have a standard definition (like self reflexive and infinitely recursive) because then we could then generate irrationals using this definition-- irrationals can only be defined by exclusion (irrationals are not rational) or (as hawking states) in terms using meta mathematics (again making computation impossible). only the similarity remains now. the similarity is probably mathematical because both (uncertainty and platonic forms of squares and circles) are basically "measurement" problems. surely the similarity has been noted by others -- not a profound observation. (incidentally, "irrationals are not rational" is also a meta mathematics statement.)

[Guest [ou...]]

Problems:

 

1. Show that

a. the square of an odd number is of the form 8k+1

b. x^4+6x^2-7 is a multiple of 128 when x is odd

 

2. Solve y^2=x^2+3 when x,y are positive integers

 

3. Simplify X=√(2+√(2+√(2+...)))

 

4. Express f(x)=1+2x+3x^2+... in closed form when |x|<1

 

Hints & Tips:

 

1. The product of 2 consecutive integers is even.

2. 3 is prime

3. Square both sides.

4. 1+x+x^2+...=1/(1-x) when |x|<1

[Guest [Kp...]]

i read up real numbers this morning (i was bored) and realized that my understanding of numbers is woefully inadequate. i was under the impression that real numbers do not exist because the infinitesimal terminates their infinite recursion but realized that real numbers cannot exist without the infinitesimal and that there is something called the archimedean property. all this is a bit frightening. i now see that it is no coincidence that cantor's diagonal is the diagonal of a square whose side is rational. again, all this is very very scary. there is something about our definition of reals that makes them uncountable. it is the symmetry -- we assume there are exactly infinite divisions between any two points... thus we get a perfect square in cantor's proof. the trouble is we have to assume this because if infinitely large or infinity exists then so does infinitely small or infinitesimal (intuition or axiomatic maybe). understanding the properties of infinitesimal (foundations of calculus/limits) and infinity, inter alia, is then necessary to understand real numbers and cantor: and i do not understand these yet.

 

i now also understand how numbers do not automatically connote to physical distance or coordinate geometry. now i am not so sure if the physical world can be mapped to real numbers. i am really glad that last night i went to bed thinking i know something about numbers and godel while today morning i can say that i actually know nothing about them because i have never gone into sufficient depth to understand the basic concept of numbers. now my decision to abandon this thought process (similarity between godel and uncertainty) is sealed because i am not capable of even understanding either, let alone comparing them for equivalence!!!!! it only remains that a similarity between godel and heisenberg is palpable to a layman like me -- but only a mathematician can study this, not me. and it is also not that the similarity is not being studied because there exists at least one paper on it -- https://www.cs.auckland.ac.nz/~cristian/HGCunpermitttedpublication.pdf but, let me leave it to the mathematicians and stop wasting my time on any sort of math or physics. i think i am more knowledgeable about benzos than numbers. so i'll stick to thinking and talking about benzos henceforth (over math or physics). needless to  say, i hereby excuse myself from this thread.

[Guest [ch...]]

kpin, if you leave then outis will be talking to himself.

 

If you're interested in the relationship between geometry and numbers, this little book is good https://www.amazon.com/Modern-Geometry-Dover-Books-Mathematics/dp/0486639622 . Shows how you can start with axiomatic geometry (points and lines) and put a metric on it (distances) to link geometry to numbers. You can construct finite or infinite geometries this way.

 

I assume you've read the Godel Escher Bach book.

 

Thanks for the link to that paper on the link between quantum uncertainty and incompleteness. Personally I think it's nonsense, but interesting nonsense

[Guest [Kp...]]

kpin, if you leave then outis will be talking to himself.

 

If you're interested in the relationship between geometry and numbers, this little book is good https://www.amazon.com/Modern-Geometry-Dover-Books-Mathematics/dp/0486639622 . Shows how you can start with axiomatic geometry (points and lines) and put a metric on it (distances) to link geometry to numbers. You can construct finite or infinite geometries this way.

 

thanks! this is precisely what i am interested in -- mapping numbers and algebra on a coordinate plane to create distances. i somehow have the feeling that all our problems are owed to this mapping -- irrational numbers are a product of this mapping. i'll order it but i am not sure when i will read it.

 

I assume you've read the Godel Escher Bach book.

 

yes i've read it. i studied godel too in comp sci in school. but i think i have understood it deeply only now.

 

Thanks for the link to that paper on the link between quantum uncertainty and incompleteness. Personally I think it's nonsense, but interesting nonsense

 

it could be nonsense. but i have decided not to comment on this subject till my knowledge on number theory is adequate. the trouble is irrational numbers are a product of the above mapping (they might exist in our mind and in nature or only in our mind). now if we are seeing irrational numbers (randomness) at a quantum level, then it means our mapping is consistent even at the quantum level (the mapping might fail at the string level -- i wonder why we are getting so many many subatomic particles in the standard model). uncertainty applies to large objects too by the way. so when you say "distances" (as you said above), you are playing with uncertainty at the visible level also (as opposed to quantum) and with mapping. i feel randomness at quantum level is being cyclically used to assert that randomness (irrationals) exist in nature. i feel randomness (same as free will) is only an irrational sequence and *might not* exist in nature. BUT i cannot say all this for sure -- i am neither qualified nor will i be in this lifetime to comment meaningfully. but, these are the nature of my confusions. when a person is confused, he gets scared. i am not trying to dramatize -- all i wish to say is that i have had enough... i want to now run away from all this confusion and divert my attention... the vertigo i get every single day, thinking about all this, is nauseating because my confusion is increasing with more knowledge every day, not decreasing.

[[al...]]

Never mind...I though that you asking if anyone was into meth.

[[Lo...]]

Never mind...I though that you asking if anyone was into meth.

 

Whew. My blood pressure just went up. I take it that this was a joke of some sort. 

[Guest [Le...]]

Never mind...I though that you asking if anyone was into meth.

 

Whew. My blood pressure just went up. I take it that this was a joke of some sort. 

 

 

[[Lo...]]

Lets see if my fried brain can remember anything:

 

(a + b)^3 = a^3 + 3*a^2*b + 3*a*b^2 + b^3

 

Now, trying to see if my memory is right:

 

(a + b)^3 =

(a + b) * (a + b) *  (a + b) =

((a + b) * (a + b)) * (a + b) =

(a^2 + a*b + b*a + b^2) * (a + b) =

(a^2 + 2*a*b + b^2) * (a + b) =

(a^3 + a^2 * b + 2*a^2*b + 2*a*b^2 + b^2*a + b^3) =

a^3 + 3*a^2*b + 3*a*b^2 + b^3

 

 

 

[[Lo...]]

I remember learning programming and writing a little program in Borland's Turbo Pascal way back in high school for calculating the number pi

 

I just googled it, but I recalled there was a series to come up with pi/4, and then I'd use that to come up with pi

 

Basically:

 

1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + 1/13 - .... = pi/4

 

Miss those days