Parallel postulate

[Philosophy Explorer]

At one time this postulate wasn't thought to be a postulate because it looked too complicated.

Now mathematicians accept this statement as a postulate (I do too). How about you? What do you think about it?

PhilX

[Philosophy Explorer]

Let me add this additional question: is Euclidean geometry as close as we get to a foundation for math?

PhilX

[Eodnhoj7]

Let me add this additional question: is Euclidean geometry as close as we get to a foundation for math?

PhilX

Yes...I would extend on it...but most of my posts argue this...I can extend further but I fear I would be beating a dead horse unless I reword it.

[Mike Strand]

Let me add this additional question: is Euclidean geometry as close as we get to a foundation for math?

PhilX

Yes...I would extend on it...but most of my posts argue this...I can extend further but I fear I would be beating a dead horse unless I reword it.

My understanding is that Euclidean geometry is one of the systems that comes from the foundations of math. Math relies on definitions of objects and concepts, axioms (assumptions, postulates) concerning those concepts, and the use of logic to infer further characteristics, statements and relationships (e.g., theorems) in and among those concepts and objects.

Simply by changing one of the postulates of Euclidean geometry (the parallel postulate -- allowing more than one line to be parallel to a given line, through a point not on the given line -- leads to curved space (non-Euclidean) geometry that's used to develop advanced physical theories (e.g., general relativity).

[Mike Strand]

By the way, PhilX, on another thread I can't find anymore -- my solution to the integral of one over the product of x and (x+1) is an inverse tangent function. My solution involves completing the square in the denominator and recognizing one of the basic integration formulas in calculus. Also a change of variable. I would still like to see your approach, if you have time.

The standard approach hinges on the fact that the derivatives of inverse trig functions turn out to be algebraic functions, like the one you presented for integration.

[Arising_uk]

At one time this postulate wasn't thought to be a postulate because it looked too complicated. …

I don't think it was because it was too complicated but that it was not clearly evident as the others were, i.e. it was not simply deducible from the other ones as the others were from each other.

Now mathematicians accept this statement as a postulate (I do too). …

What does this even mean?

How about you? What do you think about it?

I think the problems with it led to the development of other self-consistent geometries which proved that Maths is not a source of truth about reality.

[Arising_uk]

Let me add this additional question: is Euclidean geometry as close as we get to a foundation for math?

PhilX

This is a misunderstanding of the historical development of Mathematics but if there is a foundation for Maths then it is Set Theory(although no doubt wtf can put me right here as my knowledge of further mathematical developments is woefully out of date).

[wtf]

This is a misunderstanding of the historical development of Mathematics but if there is a foundation for Maths then it is Set Theory(although no doubt wtf can put me right here as my knowledge of further mathematical developments is woefully out of date).

Couldn't have said it better myself.

[Philosophy Explorer]

Arising said:

"I think the problems with it led to the development of other self-consistent geometries which proved that Maths is not a source of truth about reality."

And it doesn't have to be. What mathematicians look for is consistency. You can pick any axioms you like as long as they're consistent with one another. It's this modern viewpoint that has led to different algebras and geometries.

PhilX

[Mike Strand]

At one time this postulate wasn't thought to be a postulate because it looked too complicated. …

I don't think it was because it was too complicated but that it was not clearly evident as the others were, i.e. it was not simply deducible from the other ones as the others were from each other.

Now mathematicians accept this statement as a postulate (I do too). …

What does this even mean?

How about you? What do you think about it?

I think the problems with it led to the development of other self-consistent geometries which proved that Maths is not a source of truth about reality.

On the parallel postulate -- this came from Euclid (ancient Greece). It seemed an obvious fact, not needing proof, that if you have a perfectly flat surface (plane, which is defined by two intersecting straight lines, and you take a straight line on that plane and a point not on the line, then there could only be one straight line through that point parallel to the first line.

Try to imagine this by experimenting with a piece of paper on a table and a ruler and pencil. You can imagine one line through the point that would never cross over the first line, no matter how big the piece of paper (in your imagination) became. But any other line would be at an angle to these lines, and you can picture it eventually crossing the first line.

This seemed obvious to me as a geometry student in high school, although terms like "straight" and "flat" were taken as intuitive concepts derived from experience.

To me, it was the idea that more than one line through the point could be parallel to (never meet) the first line that was a strange idea, until I saw various models of the idea, like drawings on a saddle-shaped object. And this in turn made me question the ideas of "straight" and "flat", as other than just ideas you can picture in a class room.

As for what's addressed in the foundations of math, the following may be helpful: Here's a listing of the topics in a typical foundations course from the book, "Foundations of Higher Mathematics", by Fletcher and Patty: The Logic and Language of Proofs, Sets, Mathematical Induction, Relations and Orders, Functions, Combinatorial Proofs, Countable and Uncountable Sets, Introduction to Groups, and Foundations of Advance Calculus (sequences and convergence, subsequences and arithmetic operations on sequences, limits of functions, and continuity).

Where is geometry? Maybe under the logic and language of proofs.

[Eodnhoj7]

Let me add this additional question: is Euclidean geometry as close as we get to a foundation for math?

PhilX

Yes...I would extend on it...but most of my posts argue this...I can extend further but I fear I would be beating a dead horse unless I reword it.

My understanding is that Euclidean geometry is one of the systems that comes from the foundations of math. Math relies on definitions of objects and concepts, axioms (assumptions, postulates) concerning those concepts, and the use of logic to infer further characteristics, statements and relationships (e.g., theorems) in and among those concepts and objects.

Simply by changing one of the postulates of Euclidean geometry (the parallel postulate -- allowing more than one line to be parallel to a given line, through a point not on the given line -- leads to curved space (non-Euclidean) geometry that's used to develop advanced physical theories (e.g., general relativity).

Agreed, but it can be dually reversed in the respect that while euclidian geometry may come from the foundations of math as the statement (M → E) it may be simultaneously be observed as math coming from Euclidian Geometry as the statement (E → M). What we observe with out a shadow of a doubt it that they both reflect eachother so must extend from a third source which mediates them. The "medial" source is up for question.

However the angle from which one views the starting premise gives an entirely new definition to the predicate as one may look at the same thing from seperate angles and see seperate things.

We do understand however an inherent degree of "directional" qualites within both (which is evident in the 1d Line as a premise) and the act of quantification itself where all quantifiable empirical realities observe an inherent direct through time. For example if I quantify an orange(s) as 1 or 2 I am observe 1 or 2 projecting in one direct (at minimum) through time. In these respects quantification dependences on finiteness observes an inherent degree of directionality as time itself.

In these respects the Euclidian Line as directional takes on an equal role to the quantification of an object as time is premised in linear movement.

This nature of the "line" as a foundation for Euclidian Geometry provides a constant foundation for the fifth postulate where any "curvature" is an approximation of a line folding through itself as a series of angles conducive to a frequency....in theory.

Take for example I observe a line relative to a larger frequency (alternating angles and lines). The line exists as the line in size considering the magnitude of the angles relativistically shrinks the line. We can see an example of this in the nine point problem where 9 points in the form of a box can fit in three lines if the lines compose deep angles that extend outside the box. If I try to connect the points while remaining inside the box, the angles are not only larger but the connection requires a larger number of lines. In simpler terms, the angles through 3 lines relativistcally shrinks the 9 dot box to fix inside the angles.

So to get back on track, this "frequency" as massive angles existing relative to a line appears as a line in itself when relative to an even larger frequency...in these respects the line is always a frequence and the frequency is always a line depending on their nature of relation. Curvature would follow this similar form and function.

[Mike Strand]

My understanding is that Euclidean geometry is one of the systems that comes from the foundations of math. Math relies on definitions of objects and concepts, axioms (assumptions, postulates) concerning those concepts, and the use of logic to infer further characteristics, statements and relationships (e.g., theorems) in and among those concepts and objects.

Simply by changing one of the postulates of Euclidean geometry (the parallel postulate -- allowing more than one line to be parallel to a given line, through a point not on the given line -- leads to curved space (non-Euclidean) geometry that's used to develop advanced physical theories (e.g., general relativity).

Agreed, but it can be dually reversed in the respect that while euclidian geometry may come from the foundations of math as the statement (M → E) it may be simultaneously be observed as math coming from Euclidian Geometry as the statement (E → M). What we observe with out a shadow of a doubt it that they both reflect eachother so must extend from a third source which mediates them. The "medial" source is up for question.

Thanks, Eodnhoj7, for responding to my thoughts about math and geometry. I can only respond at this time to your "Agreed" paragraph above; the rest of your response is currently beyond my grasp.

The "medial" source, in my conception, is that of language and logic. We have language with sounds (words) having meaning or assigned meanings, statements or propositions about those words that are more or less taken at face value, and then we may use a process such as logic to obtain further statements or propositions or claims that depend on the original set of meanings and statements. So I think it's more the case that LL (language plus logic) leads to both M and E. E is now often viewed as a sub-part of M, although E developed way before many of the present-day sub-parts of M.

To make a concession to your point, however, the development of E, as early as it was, may well have informed and inspired people to develop other kinds of M!

[Eodnhoj7]

My understanding is that Euclidean geometry is one of the systems that comes from the foundations of math. Math relies on definitions of objects and concepts, axioms (assumptions, postulates) concerning those concepts, and the use of logic to infer further characteristics, statements and relationships (e.g., theorems) in and among those concepts and objects.

Simply by changing one of the postulates of Euclidean geometry (the parallel postulate -- allowing more than one line to be parallel to a given line, through a point not on the given line -- leads to curved space (non-Euclidean) geometry that's used to develop advanced physical theories (e.g., general relativity).

Agreed, but it can be dually reversed in the respect that while euclidian geometry may come from the foundations of math as the statement (M → E) it may be simultaneously be observed as math coming from Euclidian Geometry as the statement (E → M). What we observe with out a shadow of a doubt it that they both reflect eachother so must extend from a third source which mediates them. The "medial" source is up for question.

Thanks, Eodnhoj7, for responding to my thoughts about math and geometry. I can only respond at this time to your "Agreed" paragraph above; the rest of your response is currently beyond my grasp.

The "medial" source, in my conception, is that of language and logic. We have language with sounds (words) having meaning or assigned meanings, statements or propositions about those words that are more or less taken at face value, and then we may use a process such as logic to obtain further statements or propositions or claims that depend on the original set of meanings and statements. So I think it's more the case that LL (language plus logic) leads to both M and E. E is now often viewed as a sub-part of M, although E developed way before many of the present-day sub-parts of M.

To make a concession to your point, however, the development of E, as early as it was, may well have informed and inspired people to develop other kinds of M!

I can tell we are going to get along well and quick.

That is fundamentally the question though: Meaning? What not just balances out the relations of E and M, along with LL, but acts fundamentally as the "source" so to speak. Source, as point of origin, would in effect give answer to this question of meaning considering "meaning" itself can be implied as "common bond".

If we look at the root of measurements, with "arithmetic" meaning "measurement" (loosely in Greek), it is broken down to six fundamental functions, that exist as three polar duals of positive and negative, of addition/substraction, multiplication/division, power/roots.

The natures of these functions, as providing the means for quantification, observes an inherent element of empirical reality itself considering quantification is premised and proven in this.

What is observed through the senses fundamentally is change, where one empirical phenomena changes into another at different rates, with the rates of this change existing relative to other specific phenomena. Empirical phenomena are measured according to their relations with empirical phenomena, however these relations provides the boundaries of these empirical phenomena in themselves where this "change" dually acts as a constant form of mediation.

So in observing a phenomena converging with another phenomena we observe it as "addition" or "and". Such as "x" and "y" exist as "z". However this convergence observes an inherent directional quality to it, which we can observe in the nature of time itself as premised in 1 direction, where "x" and "y" observes "x" and "y" are directed towards eachother forming "z" as a new direction in time. Hence addition is positive change as unification through convergence.

We can observe the same for multiplication as "addition and addition" or "the addition of addition" where "2 x 3" = "3+3" where addition itself is quantified as a direction hence it is localized as change within change through convergence itself.

Powers follow this same manner.

One could take the premise that time as progress, or 1 dimensional movement, provides the foundation for Einstein's Relativity where up/down, forward/backward and left/right as 1 directional in themselves equate to dimensions of time. So Einstein's framework or reality is composed of six dimensions of time which exist through a seventh as time itself.

In a separate respect observing a phenomena diverging we see dually subtraction, division and roots acting in a similiar manner.

In these respects the functions, which exists through number and viceversa as a dualism of form and function where one cannot exist without the other, provide a foundation for an inherent directional quality to quantity where the nature of quantity is fundamentally rooted in spatial direction acting as a limit in itself.