Shortest distance between two points

The shortest distance between two points is never a straight line. Indeed, we can be certain that the one case that never applies to two points is that the shortest route between them is a straight line. My reasons are simple, and reasonable enough for any right-thinking person to accept without protest.

First. In the case of two separate points only, we cannot say meaningfully that any line exists between them. For all points on a line are indistinguishable from any other. Accordingly, we cannot say which point comes before or after another. And the simple and reasonable conclusion is that two points, alone, yet separate, are not separate by virtue of length, for length requires that one point come after another. We cannot then say that the shortest distance between two points is a straight line, or even a line.

Second. It would seem, from above, that there aren't any lines anywhere, for lines require that one point come after another on theor extension, yet this cannot occur on a line. Let us then re-assess our idea of a line. Let us say instead that a line is not composed of points but of positions. Now a position on a line is a construction. For example, a position could be constructed considering two points from a third point. Now this means that the shortest distance between two points is that distance upon which there is the least number of positions. We can also, instead of the term 'position' use the term 'event'. We can also drop the word 'line' because it suggests 'points'.

I come to this most simple and profound conclusion: the shortest distance between two points is defined by the least number of events, or happenings, associated with both points.

(c) John Jones

That's a complicated way of saying something completely redundant. If our "construction" or "event" is a centimeter, then the number of events is, simply said, length or distance. Your conclusion is "The shortest distance between two points is the shortest distance."

The concept of a line inherently includes the fact that it is the shortest distance between two points; that is the definition of a line. This is true a priori.

Brandon,

Your use of "prove" and "verify" is inappropriate.

What matters is the range of applicability of mathematical axioms in the modelling of physical systems.

If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.

You fail to comprehend that Mr. Jones has disproved basic Euclidean geometry that thousands and thousands of mathemeticians have verified before he was born. He intuitively, and undoubtedly with no education in the subject, sees vital aspects of the problem that they all missed for thousands of years.

Either that, or he simply likes yanking our chain.

rosborne979 wrote:

Either that, or he simply likes yanking our chain.

I'll go along with that . . . he's been attempting to yank our collective chain in a number of threads. He doesn't have staying power, though, he usually bails as soon as the heat is on.

rosborne979 wrote:

Either that, or he simply likes yanking our chain.

I'll go along with that . . . he's been attempting to yank our collective chain in a number of threads. He doesn't have staying power, though, he usually bails as soon as the heat is on.

On a flat surface, anyone can mark two points and then draw a variety of paths between them, measuring each such curve and marking down the length in each case. It soon becomes apparent that the straight line joining them is shortest, and the length of any other curve joining them is longer in proportion to how much it deviates from it.

That's a complicated way of saying something completely redundant. If our "construction" or "event" is a centimeter, then the number of events is, simply said, length or distance. Your conclusion is "The shortest distance between two points is the shortest distance."

The concept of a line inherently includes the fact that it is the shortest distance between two points; that is the definition of a line. This is true a priori.

Thalion,

Well put.

The concept of "straight" already includes the concept of "shortest distance". This raises the interesting aspect of a teleological statement applied to light which travels "such that its transit time between two points is a minimum". Thus the "rectilinear propogation" we learn about in elementary physics subsumes "Euclidean geometry" whereas Einstein (et al) were obliged to move into non-Euclidean realms where "straight" loses meaning. Alternatively "time" as in "transit time" is significantly deemed to be a "psychological construct".

It seems to me "Godels proof" may be applicable here.

Brandon,

Your use of "prove" and "verify" is inappropriate.

What matters is the range of applicability of mathematical axioms in the modelling of physical systems.

BTW Mr Jones might note that he merely shifts the problem of definition of "straight" to the definition of "event"!

fresco wrote:

Brandon,

Your use of "prove" and "verify" is inappropriate.

What matters is the range of applicability of mathematical axioms in the modelling of physical systems.

No, actually they are not. This is one of Euclid's posulates. Mr. Jones fancies that he has disproven it, but generations of mathematicians have verified that it appears to be an axiom that successfully models physical reality.

In Euclidean space this is consequence of the 5th postulate, e.g.'

Quote:

If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.

This postulate is equivalent to what is known as the parallel postulate.

In planar space this can be used to demonstrate the shortest distance between two points is a straight line theorem. If you reject this postulate, you exit Euclidean (parabolic) geometry for either hyperbolic or elliptic geometry, both of which have a different solution for this ?'shortest distance problem.

In two dimensions hyperbolic analytic geometry is expressed as Poincaré hyperbolic disk. Similarly, in two dimensions elliptical geometry is expressed as Spherical Geometry.

Both of these alternate geometries provide a practical solution for the shortest path between point problems.

Nevertheless, Mr. Jones revelation has been extensively developed well prior to his (& my) birth.

Rap c∫;?/

I fancy that the shortest distance between two points as demonstrated by the 5th postulate is not of a line, but of the least number of events associated with both points.