What is your difinition of truth?

[Citizen James]

there is only one truth.  Truth only has one side to it, something is either true or it's false.  There's nothing in between.

For example: it would be total bs if I said the phrase "I see yellow flowers" and you said that's true for you and not for me.

Why?  It has to do with the language.  I perceive the flowers to be yellow.  Now you may not percieve the flowers to be yellow, yourself, but you cannot deny the fact that I see them as yellow.

This statement is a lie.


Please tell me whether you catagorize the previous statement as true or false?

[Gabu]

This statement is a lie.


Please tell me whether you catagorize the previous statement as true or false?

There's actually a branch of logic that deals with statements such as the one above, and deems them "meaningless statements": they're neither true nor false, and they don't even really mean anything at all.

Personally, I don't think that contradicts with what I said.  I asserted that something either is something or it is nothing, and that statement is, well, nothing.

[○∙◄☻¥tπ[╪AV┼cVê└]

As an aside, I personally take a special case of Euler's equation, which reads as follows:

e^iπ + 1 = 0

...to be my personal proof of the existence of God.

Gabu, I am impressed with your posts in this thread, especially since my career in mathematics ended after my first  course in college calculus.

Could you enlighten this layman on the above equation and your take on it? (Don't work too hard, please - I don't expect more than slight enlightenment.)

Eulers formula is e^(it) = cos(t) + i sin(t), where i=sqrt(-1)
It can be derived from the Taylor series of cos(x), sin(x), and e^x.
When you plug in t=Pi, you get
e^(i Pi) =  cos(Pi) + i sin(Pi) = -1.

[DanielX]

The truth is what is.

[Gabu]

The truth is what is.

But what if it isn't?   </randomMeaninglessPhilosophy>

[Mort from NewYawk]

Could you enlighten this layman on the above equation and your take on it? (Don't work too hard, please - I don't expect more than slight enlightenment.)

I'd be happy to, except I'm not 100% sure what you're asking for.  Do you want what I find amazing about the equation I listed?  What Euler's equation is?  Where it comes from?  What its relevance to reality is?

All of these I can probably answer to some extent, though some may be kind of tricky.

What you find amazing about the equation you listed.

[muon2]

The Soult Lexicon

Fact: 1) That which is undisputable. 2) That which can be proven through direct empirical analysis.

Truth: 1) A universal principle that can be derived at from philosophical reflection, but not proven by direct imperical evidence. 2) That which is derived at through the personal expiriences and thoughtful reflection of an individual or group of individuals.

Soulty makes a very useful distinction here.

Facts (or observations) are verifiable empirical statements. The trajectory of a ball in flight falls into this category. One can talk about mathematical transformations to explain the difference in viewpoints of facts. This can be limiting when one deals in quantum events that do not transform based on the observer, but must succumb to a statistical interpretation.

Truths are the fundamental principles into which we fit our facts. In physics some example might include:

An observer who is not accelerating (changing speed or direction)cannot absolutely determine their state of motion.

An observer who is accelerating cannot distinguish their acceleration from a gravitational force.

The speed of light in a vacuum is the same to all observers.

A particle has no definite intrinsic value for dynamical variables such as position or momentum, but has a value set at the instant of an observation.

[muon2]

As an aside, I personally take a special case of Euler's equation, which reads as follows:

e^iπ + 1 = 0

...to be my personal proof of the existence of God.

Gabu, I am impressed with your posts in this thread, especially since my career in mathematics ended after my first  course in college calculus.

Could you enlighten this layman on the above equation and your take on it? (Don't work too hard, please - I don't expect more than slight enlightenment.)

One of the things that makes this equation so elegant is that it ties together fundamental symbols from different branches of mathematics into a single expression.

In arithmetic the basic unit of counting is 1.

In group theory 0 and 1 are the identity elements for addition and multiplication respectively. That is if you add 0 you get the same result, and if you multiply by 1 you get the same result.

In algebra the symbol i is required to provide solutions to all quadratic equations. The simplest example is to solve the equation x² = -1. The answer is the "imaginary" unit i or -i.

In geometry the symbol π represents the ratio of the circumference of the circle to its diameter. It has been known for a long time that this ratio is not derivable from simple algebra.

In calculus the symbol e is the base for the natural logarithm. The expression e^x is the only function (other than 0) whose derivative is itself.

[David S]

As an aside, I personally take a special case of Euler's equation, which reads as follows:

e^iπ + 1 = 0

...to be my personal proof of the existence of God.

Would you elaborate on that? My recollection of Euler's equation is that it can be used to predict the load at which a column will buckle. Although my geezer brain is really stretching to remember that far back.

[Beet]

As an aside, I personally take a special case of Euler's equation, which reads as follows:

e^iπ + 1 = 0

...to be my personal proof of the existence of God.

Gabu, I am impressed with your posts in this thread, especially since my career in mathematics ended after my first  course in college calculus.

Could you enlighten this layman on the above equation and your take on it? (Don't work too hard, please - I don't expect more than slight enlightenment.)

One of the things that makes this equation so elegant is that it ties together fundamental symbols from different branches of mathematics into a single expression.

In arithmetic the basic unit of counting is 1.

In group theory 0 and 1 are the identity elements for addition and multiplication respectively. That is if you add 0 you get the same result, and if you multiply by 1 you get the same result.

In algebra the symbol i is required to provide solutions to all quadratic equations. The simplest example is to solve the equation x² = -1. The answer is the "imaginary" unit i or -i.

In geometry the symbol π represents the ratio of the circumference of the circle to its diameter. It has been known for a long time that this ratio is not derivable from simple algebra.

In calculus the symbol e is the base for the natural logarithm. The expression e^x is the only function (other than 0) whose derivative is itself.

Interesting!

[David S]

As an aside, I personally take a special case of Euler's equation, which reads as follows:

e^iπ + 1 = 0

...to be my personal proof of the existence of God.

Would you elaborate on that? My recollection of Euler's equation is that it can be used to predict the load at which a column will buckle. Although my geezer brain is really stretching to remember that far back.

BTW for engineers; Euler's equation is not used by engineers for determining column buckling loads. It is a theoretical equation which gives answers on high side i.e. unsafe. Don't use it for that.

[Gabu]

As an aside, I personally take a special case of Euler's equation, which reads as follows:

e^{i π} + 1 = 0

...to be my personal proof of the existence of God.

Would you elaborate on that? My recollection of Euler's equation is that it can be used to predict the load at which a column will buckle. Although my geezer brain is really stretching to remember that far back.

Well, the general form of Euler's equation looks like this:

e^{i θ} = cos θ + i sin θ

In all of my mathematical career, I've only actually seen one practical application of it so far, which is to solve a differential equation of the form

m x''(t) + k x(t) = 0

for m, k > 0.

Some who enjoy physics will likely recognize this as the case where a mass is oscillating on a spring.  Its solutions are

x₁(t) = e^{i sqrt(k/m)}
x₂(t) = e^{-i sqrt(k/m)}

and then you can use Euler's equation as well as a property of second-order differential equations* to reduce this down to cosines and sines in the real numbers to show that the solution is indeed oscillatory.

The reason that I personally find the special case so fascinating, however, is because of what muon2 noted, which is that it essentially ties together all of the fundamental numbers in mathematics in one tiny, compact formula.

In addition, just look at it: you take an irrational number, raise it to the power of another irrational number multiplied by an imaginary number, and you get an integer.  Personally, I think that's wild.

*The property in question is that, if x₁ and x₂ are both solutions to a differential equation, then so is x₃ = c₁ x₁ + c₂ x₂, with c₁, c₂ constants.

Using Euler's equation on the two solutions above, we get

x₁(t) = cos(sqrt(k/m)) + i sin(sqrt(k/m))
x₂(t) = cos(-sqrt(k/m)) + i sin(-sqrt(k/m))

For x₂, we now note that, since cos is an even function and sin is an odd function, we have

x₂(t) = cos(sqrt(k/m)) - i sin (sqrt(k/m))

since a property of an even function f and an odd function g is that f(-x) = f(x) and that g(-x) = -g(x)

Now we use the property of second-order differential equations above to get the following other solutions to this equation:

x₃(t) = 1/2 x₁(t) + 1/2 x₂(t)
x₃(t) = 1/2 cos(sqrt(k/m)) + 1/2 i sin(sqrt(k/m)) + 1/2 cos(sqrt(k/m)) - 1/2 i sin(sqrt(k/m))
x₃(t) = cos(sqrt(k/m))

x₄(t) = -1/2 i x₁(t) + 1/2 i x₂(t)
x₄(t) = -1/2 i cos(sqrt(k/m)) - 1/2 i² sin(sqrt(k/m)) + 1/2 i cos(sqrt(k/m)) - 1/2 i² sin(sqrt(k/m))
x₄(t) = -1/2 (-1) sin(sqrt(k/m)) - 1/2 (-1) sin(sqrt(k/m))
x₄(t) = 1/2 sin(sqrt(k/m)) + 1/2 sin(sqrt(k/m))
x₄(t) = sin(sqrt(k/m))

And then we can finally combine x₃(t) and x₄(t) together to get our final general solution:

x(t) = C₁ cos(sqrt(k/m)) + C₂ sin(sqrt(k/m))

which is indeed an oscillatory function in the real numbers.

[muon2]

As an aside, I personally take a special case of Euler's equation, which reads as follows:

e^{i π} + 1 = 0

...to be my personal proof of the existence of God.

Would you elaborate on that? My recollection of Euler's equation is that it can be used to predict the load at which a column will buckle. Although my geezer brain is really stretching to remember that far back.

Well, the general form of Euler's equation looks like this:

e^{i θ} = cos θ + i sin θ

In all of my mathematical career, I've only actually seen one practical application of it so far, which is to solve a differential equation of the form

m x''(t) + k x(t) = 0

for m, k > 0.

Some who enjoy physics will likely recognize this as the case where a mass is oscillating on a spring.  Its solutions are

x₁(t) = e^{i sqrt(k/m)}
x₂(t) = e^{-i sqrt(k/m)}

and then you can use Euler's equation as well as a property of second-order differential equations* to reduce this down to cosines and sines in the real numbers to show that the solution is indeed oscillatory.

Oscillatory problems are the heart and soul of physics. The complex notation from the equation is used in many settings. Electricity and magnetism, electronics, and quantum mechanics are a few areas where complex notation is a valuable tool.

[Storebought]

My definition of truth is anything that Britney Spears tells me on her new MTV show 'Chaotic'

Seriously, I lean toward the realist notion of truth: It is a positive thing that can be observed in nature or deduced from experience.