Does the square root of negative one lead to a contradiction?

[zaphod]

That's not false. What you have written above here, is actually true.

 

but the fact of the matter is that there are no contradictions involved.

[Johnny5]

but the fact of the matter is that there are no contradictions involved.

 

 

Regarding irrational numbers, or regarding complex numbers?

[Dave]

Regarding irrational numbers, or regarding complex numbers?

 

Both, I would hope

 

One other point I would like to make (that's already been made by people here) is that contradiction is rather the wrong word to be using. Something like "Does defining the complex numbers lead to inconsistencies in mathematics?" would be a better way of phrasing your question.

[Johnny5]

Both' date=' I would hope

 

One other point I would like to make (that's already been made by people here) is that contradiction is rather the wrong word to be using. Something like "Does defining the complex numbers lead to inconsistencies in mathematics?" would be a better way of phrasing your question.[/quote']

 

I agree.

 

I suppose, really it goes to the meaning of the mathematical expression.

 

What does the following mathematical sentence mean?????

 

E.G.

 

 

x^2-9 = 0

 

We can factor the LHS, something we cannot do to sentences in general, to get here:

 

(x+3)(x-3)=0

 

if x=-3 then sentence above is true

if x=3 then sentence above is true

 

if not (x=-3) and not(x=3) then sentence above is false.

 

Hence, x is a variable, something which makes the sentence not really a statement at all.

 

it isn't until after you instantiate the variable with a symbol which denotes a number, that you end up with a statement, which may either be true or false, as the case may be. (but of course this leads to 'free variable' 'bound variable' nonsense present in first order logic, and I think it's wise to stay away from those terms)

 

As far as whether you want to say 'inconsisteny' or 'contradiction' i think the fact remains, which is that we ourselves are performing mental operations, and we don't want our 'rules' to be inconsistent with one another.

 

We certainly treat such expressions as having truth value, and manipulate them as though they are statements with meaning.

 

Like i said, I just want to read what others think about it.

 

Regards

 

PS: And not to lose sight of the main point...

 

if i<0 then problem

if i=0 then problem

if i>0 then problem

[zaphod]

And not to lose sight of the main point...

 

if i<0 then problem

if i=0 then problem

if i>0 then problem

 

 

[Dave]

 

Well, quite, That last thing made pretty much no sense at all

[kriminal99]

which logic is being redifined here? i dont understand.

 

if you take a set of objects and perform certain operations on them' date=' sometimes you will realize that the set isnt closed under that operation.

 

for example, when the greeks tried to take the square root of 2, they couldnt fathom the existence of a number not part of the rational numbers. do the so called "irrational" numbers lead to a contradiction? i would hope to think not.

 

so when you've got the idea of the square root of a negative number, the solution falls outside the original set of numbers. so just as math was plunged into irrational numbers to find solutions to square roots of "non-square" numbers, the "imaginary" number was created and the complex number system was used to find solutions to square roots of negative numbers.

 

"irrational" and "imaginary" arent meant to be taken literally.

 

they're perfectly valid number systems which allow calculations to be made.

 

just because you cant have 3[i']i[/i] apples doesnt mean that the number system is a fallacy. i mean you cant have pi apples either (no matter how hard you try).

 

I figure what the greeks probably couldn't fathom (and with good reason) is how the rational numbers could be dense and yet not contain all possible numerical values. That is a logical contradiction.

 

Of course irrational "numbers" aren't really numbers at all they are just non-mathematical infinite algorithms for creating numbers, motivated by the need to measure lengths in directions they already shrunk into infinitely small points in their primitive coordinate system (or similar needs)... Even if there was no alternative to this system, there would be no need to call them numbers since practically they are only ever treated as rationals - the result of said algorithm after following it to a certain point where the decimal of the precision you want no longer changes...

 

What I am arguing here is that mathematicians, to have to get around their contradictions, end up having to make statements like "a number isn't really a number its kinda like a number but has this that and the other attributes instead" and other such gibberish. In whatever warped world this eventually puts you in, it might even be said that their belief set is coherent, but they are still contradicting common terms and getting farther and farther away from being able to claim that math is in any way motivated by reality. (Which eventually would make mathematicians religious fanatics as opposed to bastions of reason)

 

I can't have pi apples because theres no such thing (in the real world) as a perfect circle, perhaps because theres no such thing as an infinitely small length... and therefore no such thing as a ratio of a circle's circumference to its diameter...

 

@ Matt

 

IMO The numbers that actually exist are the natural numbers alone. We have no reason to believe that anything exists in infinitely small quantities. Without those, any fraction is really a discreet natural number system on a different scale. But then we might use rational numbers to avoid scales where we have like 2000000000 molecules of water for a subjectively small amount of water. IMO Irrational numbers are not numbers, but rather as described above. Once defined to a certain precision they reach the point of discreet units (and decimal places of the irrational number past that point become irrelevant), and if we knew how far they could be broken down on the original scale then they could just be considered rational in that scale.

 

If a logical calculus is effectively created (I know what types of problems have been run into trying this and why) then "irrational numbers" might eventually be considered objects of this system rather than numbers, and numbers will be left as simple quantities.

[matt grime]

But that only is a contradiction if you claim that numbers are in some sense real. We do not. So we aren't making any contradictions. You are misrepresenting mathematicians (or this one at least) if you say we are.

 

A very famous mathematician once said that "God invented the whole numbers, the rest is the work of man", and I agree. But there isn't 'a contradiction because we aren't saying numbers (real or complex or anything) are in anyway representative of anything in the real world.

 

We freely admit that the things we do may, indeed almost certainly, have no realization in the real world. How can that be religious fanaticism if we admit our "god" is fictional? All we say is that some parts of what we do are useful for modelling the real world, but the models aren't necessarily themselves real.

 

If you don't think there is a real life case where -1 has a square root then you should read up on electricl engineering, or qunatum physics (including its experimentally verified parts).

 

Please stop accusing us of making claims that we do not make. We can even demonstrate to you that almost every real number is not constructible for heaven's sake, so how can we be making any contradictory statements? All that is happening here is you are using a different definition for number than a mathematician might - though in rigourous terms we would never say number alone, but qaulify it with real or rational or some such. And please don't think that because of the unfortunate naming of them as real and imaginary this in anyway reflects our opinion of their eixtence in any sense.

 

The naturals may be said to exist, 0 is to many people doubtful but to a mathematician it is simply adding in an identity to a semigroup, then adding formal inverses gives the integers, localization gives the rationals, completion gives the reals and closure yields the complex numbers. It is because of the exact issues you raise that mathematicians have devised this to be on a formal and concrete footing *within the realms of mathematics* - we have definitely divorced this from reality a long time ago, and we don't claim otherwise!

[kriminal99]

But then you are doing something dishonest. You are depending on errors in the way people think and the weaknesses of induction to cause people to have faith in your "non reality founded" belief set.

 

Anotherwords on day 1 of math class you tell them 1 +1 = 2. They see this is true in their every day life. Then you tell them a bunch of other stuff they can see in their life. Then you come up with all these random terms and ways of thinking that are not optimized for practicality or based on reality. But because of the way people think they are reluctant to challenge your claims... Even though the stuff you told them that was helpful could have been part of many mathematical belief sets. It also might have been MADE by someone who would follow another belief set because that person believed it was somehow better, and you may just be using it in your belief set to decieve people into thinking you are an authority so they won't doubt you. (You isn't pointing to any one person exactly)

 

Even as you say we are not claiming any connection to reality, you are relying on people to respect you as a mathematician, which is dependent on people's belief that what you have to say is the not only useful, but the best way of thinking about it.

 

If you are constantly searching for ways to optimize and revise the mathematical belief set, then you are doing the best you can and there can be nothing wrong with that. Even if every mathematician does not do so but some sort of specialize in it, and then the rest just remain open to these ideas and consider their worth then there can be nothing wrong with the situation. However if you, for selfish emotional reasons, reject all alternative ideas, and fail to investigate new ways of thinking, then what you are doing is basically brainwashing.

 

The issue I have with religious groups is their methods of persuasion. If youve attended any religious sermon before you probably know that the majority of their arguments are metaphors of one form or another, and they make no argument to connect the two situations being related in the metaphor. Anotherwords, they are circumventing each person's faculty of reason in order to get them to submit to their views. In this sense mathematicians are behaving in the same way.

 

Of course if one day people become more aware of their weaknesses and how to get around them this will become less of an issue. The only ideas which will ever gain any social signifigance are those which are perfectly logical. I am working towards this myself, but I have no way of knowing if I or anyone else will ever be succesful in this.

 

The cases where -1 "has" a square root is no doubt a result of an earlier lack of connection between mathematics and reality. Anotherwords for example if you try to model something in real life space where logicaly there can be no reason why the height of a location from an imaginary axis can not have a square root despite the fact it is behind wherever you put the imaginary x-axis. So if you want to keep thinking the way you have been of course you have to make up ideas like sqrt(-1) (which may in some way contradict themselves) in order for your belief set to be able to accoplish something in reality.

 

EDIT: When I was talking about contradictory statements, as I said before the belief set may be coherent. (Meaning if you think about it a certain way, the statements are not contradictory) However when you fail to make a complete connection to reality, but you still want to accomplish certain things in reality using your belief set, you end up having to make silly statements as I mentioned before, or at best end up having to redefine common terms to suit your needs. Also if you begin changing things in a belief set to suit any need other than consistency, then you are going to end up with contradictions even within your belief set. If the math belief set was perfectly grounded in reality, then the need to accoplish things in the real world and the need for consistency would be one and the same.

[matt grime]

Brainwashing? Erm, no. We simply state that if we accept these axioms and those rules of logical inference then the following deductions are true, these statements are false, and the system may or may not be incomplete (if it models the natural numbers and is finitely axiomatized then it cannot be complete - Goedel).

 

In what way is that brainwashing or remotely religious? Different parts of mathematics start from different axioms and some even have different rules of logic, but we all admit before hand what they are.

 

Do not confuse the necessary eliding of details in highschool, where the subtleties are omitted and wouldn't be understood, with any attempt at dishonesty.

 

Sadly some people become fixed and think the "baby" explanations are absolutely true.

 

To be honest your post does indicate a lack of exposure to higher level mathematics, and indeed the concepts of axioms etc.

 

We are never anything but honest about axiomatic systems. Why all this antipathy and refusal to accept mathematics simply for what it is; it doesn't pretend to be anything of the things you appear to claim it is.

[Johnny5]

If I may, i think i know what might be constructive.

 

According to history, the mathematician Gauss proved that every integral rational equation of degree n, has exactly n roots.

 

Apparently, so the history books say, one of his proofs was his dissertation for his doctorate. He gave three others.

 

Now, I have never even seen one of these supposed proofs.

 

So i challenge anyone, to produce a proof of that which is italicized above. If such a proof is successful, it must be the case that:

 

x^2+1=0

 

has exactly two roots, and they will be +i, and -i.

 

So if the proof doesn't contain any errors, then imaginary numbers will be justified.

 

I suspect though, that there is a logical error in all four of Gauss' proofs.

[Dave]

So i challenge anyone' date=' to produce a proof of that which is italicized above. If such a proof is successful, it must be the case that:

 

x^2+1=0

 

has exactly two roots, and they will be +i, and -i.

 

So if the proof doesn't contain any errors, then imaginary numbers will be justified.[/quote']

 

I very much suspect that this proof would already assume existance of the complex numbers, and as such it doesn't justify very much at all.

 

I suspect though, that there is a logical error in all four of Gauss' proofs.

 

I highly doubt this.

 

As matt has said about twice now, we can do whatever we want to in mathematics as long as it makes logical sense. It makes perfect sense to define complex numbers. If you don't like it, then that's your problem, not ours.

[Johnny5]

I very much suspect that this proof would already assume existance of the complex numbers' date=' and as such it doesn't justify very much at all.

[/quote']

 

I would like to see said proof. Do you know it?

 

Kind regards

[Dave]

No, I don't. I daresay it's quite long though, so I won't even try to attempt to replicate it here. I doubt I would understand it anyway

[zaphod]

 

I suspect though' date=' that there is a logical error in all four of Gauss' proofs.[/quote']

 

well how about you take a few math classes, then get back to us with your contradictions to the proofs of the Fundamental Theorem of Algebra (which states that "Every polynomial equation of degree n with complex coefficients has n roots in the complex numbers.")

 

jesus. you say you've never seen the proofs, but you automatically suspect that they're wrong. this alone is should be a good indicator of how much time we're all wasting in this thread.

 

and here, http://www.google.ca/search?hl=en&q=proof+fundamental+theorem+algebra&btnG=Search&meta=

 

come back when you've got your contradictions.

[kriminal99]

Brainwashing? Erm' date=' no. We simply state that if we accept these axioms and those rules of logical inference then the following deductions are true, these statements are false, and the system may or may not be incomplete (if it models the natural numbers and is finitely axiomatized then it cannot be complete - Goedel).

 

In what way is that brainwashing or remotely religious? Different parts of mathematics start from different axioms and some even have different rules of logic, but we all admit before hand what they are.

 

Do not confuse the necessary eliding of details in highschool, where the subtleties are omitted and wouldn't be understood, with any attempt at dishonesty.

 

Sadly some people become fixed and think the "baby" explanations are absolutely true.

 

To be honest your post does indicate a lack of exposure to higher level mathematics, and indeed the concepts of axioms etc.

 

We are never anything but honest about axiomatic systems. Why all this antipathy and refusal to accept mathematics simply for what it is; it doesn't pretend to be anything of the things you appear to claim it is.[/quote']

 

You missed the entire point of that post...

 

If there were any truth to the statement "your post does not indicate a lack of exposure to higher level mathematics" then you would be debating one of the arguments I made directly rather than completely dodging them and making false statements to try and decieve people into believing you. This is a good example of intellectual dishonesty or attempted brainwashing.

[Johnny5]

well how about you take a few math classes' date=' then get back to us with your contradictions to the proofs of the Fundamental Theorem of Algebra (which states that "Every polynomial equation of degree n with complex coefficients has n roots in the complex numbers.")

 

jesus. you say you've never seen the proofs, but you automatically suspect that they're wrong. this alone is should be a good indicator of how much time we're all wasting in this thread.

 

and here, http://www.google.ca/search?hl=en&q=proof+fundamental+theorem+algebra&btnG=Search&meta=

 

come back when you've got your contradictions.[/quote']

 

Ok this post really does it.

 

Here is what I will now do.

 

I will utilize all of my powers of concentration, all of my skill in reasoning, and access any and all of the General order spatiotemporal modal binary deontic doxastic logic which i can, and prove unequivocally that it is not the case that every algebraic equation of degree n, has n solutions.

 

Give me one day.

[Tom Mattson]

You don't even need a proof, all you need to do is come up with a single counterexample. Can't wait to see it.

[matt grime]

You missed the entire point of that post...

 

If there were any truth to the statement "your post does not indicate a lack of exposure to higher level mathematics" then you would be debating one of the arguments I made directly rather than completely dodging them and making false statements to try and decieve people into believing you. This is a good example of intellectual dishonesty or attempted brainwashing.

 

 

What arguments? You've made no mathematical statements at all, only offered some opinions as to how we're brainwashing people with contradictions. Evidently you are ignorant of mathematics. Feel free to post a list of your qualifications in mathematics.

 

In fact I have absolutely agreed with the statement that the only "numbers" that may in any reasonable sense be said to exist in anyway in the real world are the natural numbers, and that the rest are "inventions" of mathematicians. That doesn't stop us reasoning about them any more than the fact that Hamlet is fictional stop us reasoning that he has some issues with his parents. The difference of course being that mathematical statements are not subjectively true, but nothing is absolutely true. Do you think, for instance that "through every point not lying on a given line there passes a unique parallel line"? If you say yes, absolutely then you truly are ignorant of mathematics.

 

This doesn't stop statements in mathematics being true within the carefully defined parameters that we offer, no does it diminish the uses of mathematics.

[Johnny5]

You don't even need a proof, all you need to do is come up with a single counterexample. Can't wait to see it.

 

I already had one in mind Tom, or Thomas.

 

 

 

(x-2)(x-2)=x^2-2x-2x+4=x^2-4x+4

 

 

Start with:

 

x^2 - 4x +4 = 0

 

The degree is 2. By the meaning of two, there are two roots.

 

The only root is 2. So there is only one root.

 

Thus contradicting the hypothesis that there are two roots.

 

QED

 

Did you see this coming?

 

Regards

 

PS: by the way i didn't use all that logical power to find a counterexample.

[matt grime]

Oh, for pity's sake Johnny, learn about repeated roots, or roots "with multiplicity" and stop with these childish postings. By your own reasoning it has two linear factors.

[Johnny5]

Oh, for pity's sake Johnny, learn about repeated roots, or roots "with multiplicity" and stop with these childish postings. By your own reasoning it has two linear factors.

 

But you say 'roots' in the plural Matthew.

 

yet there can be only one.

 

x^2-4x+4=0

 

I don't think it's childish by the way. I am serious, there is only one root.

 

 

Draw an X axis, and a Y axis, plot the graph of

 

x^2-4x+4

 

ok let that have been done.

 

So here is the function which you have plotted on Maple, or Mathematica, or what have you:

 

y(x) = x^2-4x+4

 

Now, look at the plot and tell me how many times the curve intersects the x axis?

 

One or two?

[Tom Mattson]

Did you see this coming?

 

No' date=' but only because I didn't think that you were quite this daft. I stand corrected.

 

PS: by the way i didn't use all that logical power to find a counterexample.

 

You don't say.

[matt grime]

the set of roots contains one element, but the poly splits into two linear factors as Gauss's theorem states. Learn the sodding theorem properly first.

[Johnny5]

This is very funny. Why don't you prove the theorem Matt?

 

You are the professional mathematician, not I.