Intuition and Logic in Mathematics by Henri Poincaré

On the other hand, look at Professor Klein: he is studying one of the most abstract questions of the theory of functions to determine whether on a given Riemann surface there always exists a function admitting of given singularities. What does the celebrated German geometer do? He replaces his Riemann surface by a metallic surface whose electric conductivity varies according to certain laws. He connects two of its points with the two poles of a battery. The current, says he, must pass, and the distribution of this current on the surface will define a function whose singularities will be precisely those called for by the enunciation.

It is necessary to see the stance Poincaré had in relation to Klein, and to see how this is being played out today. While I write here I see where such thinking moved from a fifth dimensional perspective, has been taken down “to two” as well as” the thinking about this metal sheet.

So I expound on the virtues of what Poincaré saw, versus what Klein himself was extrapolating according to Poincaré’s views. We have results today in our theoretics that can be describe in relation. This did not take ten years of equitation’s, but a single picture in relation.

Graduating the Sphere

Imagine indeed a inductive/deductive stance to the “evolution of this space” around us. That some Klein bottle “may be” the turning of the “inside out” of our abstractness, to see nature is endowably attached to the inside, that we may fine it hard to differentiate.I give a example of this In a link below.

While I demonstrate this division between the inner and outer it is with some hope that one will be able to deduce what value is place about the difficulties such a line may be drawn on this circle that we may say how difficult indeed to separate that division from what is inside is outside. What line is before what line.

IN “liminocentric structure” such a topology change is pointed out. JohnG you may get the sense of this? All the time, a psychology is playing out, and what shall we assign these mental things when related to the sound? Related to the gravity of our situations?

You see, if I demonstrate what exists within our mental framework, what value this if it cannot be seen on the very outskirts of our being. It “cannot be measured” in how the intellect is not only part of the “sphere of our influence” but intermingles with our reason and emotive conduct. It becomes part of the very functions of the abstractness of our world, is really set in the “analogies” that sound may of been proposed here. I attach colour “later on” in how Gravity links us to our world.

I call it this for now, while we continued to “push for meaning” about the nature of space and time.

Here’s an analogy to understand this: imagine that our universe is a two-dimensional pool table, which you look down on from the third spatial dimension. When the billiard balls collide on the table, they scatter into new trajectories across the surface. But we also hear the click of sound as they impact: that’s collision energy being radiated into a third dimension above and beyond the surface. In this picture, the billiard balls are like protons and neutrons, and the sound wave behaves like the graviton.

While of course I highlighted an example of the geometer in their visual capabilities, it is with nature that such examples are highlighted. Such abstractness takes on “new meaning” and settles to home, the understanding of all that will ensue.

The theorems of projective geometry are automatically valid theorems of Euclidean geometry. We say that topological geometry is more abstract than projective geometry which is turn is more abstract than Euclidean geometry.

It is of course with this understanding that “all the geometries” following from one another, that we can say that such geometries are indeed progressive. This is how I see the move to “non-euclidean” that certain principals had to be endowed in mind to see that “curvatures” not only existed in the nature of space and time, that it also is revealed in the Gaussian abstracts of arcs and such.

These are not just fixations untouched abstractors that we say they have no home in mind, yet are further expounded upon as we set to move your perceptions beyond just the paper and thought of the math alone in ones mind.

It is my opinion that in teaching it is not only admissible, but absolutely necessary, to be less abstract at the start, to have constant regard to the applications, and to refer to the refinements only gradually as the student becomes able to understand them. This is, of course, nothing but a universal pedagogical principle to be observed in all mathematical instruction ….

I am led to these remarks by the consciousness of growing danger in Germany of a separation between abstract mathematical science and its scientific and technical applications. Such separation can only be deplored, for it would necessarily be followed by shallowness on the side of the applied sciences, and by isolation on the part of pure mathematics ….

Felix Christian Klein (April 25, 1849 – June 22, 1925) was a German mathematician, known for his work in group theory, function theory, non-Euclidean geometry, and on the connections between geometry and group theory. His 1872 Erlangen Program, classifying geometries by their underlying symmetry groups, was a hugely influential synthesis of much of the mathematics of the day.

See:

Having a picture in mind is helpful along with number patterns (like the Pascal Triangle). There’s the picture for the expanding universe (we on the real number surface of an expanding balloon with complex numbers inside)… many-worlds, Feynman paths, string theory worldsheets all kind of have a branching with loops picture and you can sort of integrate over the sheet between points to get relative probabilities or something like that (the pictures are never perfect). Liminocentric pictures work in lots of places; gravity, particles, string theory dualities, anything with the big equals small, beginning equals end, outside equals inside.

Having a picture in mind is helpful along with number patterns (like the Pascal Triangle). There\’s the picture for the expanding universe (we on the real number surface of an expanding balloon with complex numbers inside)… many-worlds, Feynman paths, string theory worldsheets all kind of have a branching with loops picture and you can sort of integrate over the sheet between points to get relative probabilities or something like that (the pictures are never perfect). Liminocentric pictures work in lots of places; gravity, particles, string theory dualities, anything with the big equals small, beginning equals end, outside equals inside.

JohnG:(the pictures are never perfect).If conceived as a series of ever-wider experiential contexts, nested one within the other like a set of Chinese boxes, consciousness can be thought of as wrapping back around on itself in such a way that the outermost ‘context’ is indistinguishable from the innermost ‘content’ – a structure for which we coined the term ‘liminocentric’.For me this would be an interesting exercise on what description, if drawn/painted, would express “the nature of the Liminocentric structure.” A kaleidoscopic pattern perhaps?:)”…underwriting the form languages of ever more domains of mathematics is a set of deep patterns which not only offer access to a kind of ideality that Plato claimed to see the universe as created with in the Timaeus; more than this, the realm of Platonic forms is itself subsumed in this new set of design elements– and their most general instances are not the regular solids, but crystallographic reflection groups. You know, those things the non-professionals call . . . kaleidoscopes! * (In the next exciting episode, we’ll see how Derrida claims mathematics is the key to freeing us from ‘logocentrism’– then ask him why, then, he jettisoned the deepest structures of mathematical patterning just to make his name…)* H. S. M. Coxeter, Regular Polytopes (New York: Dover, 1973) is the great classic text by a great creative force in this beautiful area of geometry (A polytope is an n-dimensional analog of a polygon or polyhedron. Chapter V of this book is entitled ‘The Kaleidoscope’….)”While E8 is very complex mathematically and takes up a lot of storage space in terms of memory on a computer, it would seem that such an expression of the complexity of nature “could manifest” as the “individual mind” using this pattern?:)”Colour and gravity” would remove the distinctiveness of the lines in that pattern, yet, could it be said, that a flower most appropriately is an expression of every soul?:)The intellect and the emotive states mix(entangle) and while these themselves are reflective of an inner thinking world, they are also are reflected in the world around us.Line up the world inside, overlap the schematics, integrate the emotion, and what may come about, may be quite a buzz:) Maybe an alignment of a kind that connects all three. The emotive, the intellect and the spiritual.Best,

JohnG:(the pictures are never perfect).If conceived as a series of ever-wider experiential contexts, nested one within the other like a set of Chinese boxes, consciousness can be thought of as wrapping back around on itself in such a way that the outermost \’context\’ is indistinguishable from the innermost \’content\’ – a structure for which we coined the term \’liminocentric\’.For me this would be an interesting exercise on what description, if drawn/painted, would express \”the nature of the Liminocentric structure.\” A kaleidoscopic pattern perhaps?:)\”…underwriting the form languages of ever more domains of mathematics is a set of deep patterns which not only offer access to a kind of ideality that Plato claimed to see the universe as created with in the Timaeus; more than this, the realm of Platonic forms is itself subsumed in this new set of design elements– and their most general instances are not the regular solids, but crystallographic reflection groups. You know, those things the non-professionals call . . . kaleidoscopes! * (In the next exciting episode, we\’ll see how Derrida claims mathematics is the key to freeing us from \’logocentrism\’– then ask him why, then, he jettisoned the deepest structures of mathematical patterning just to make his name…)*While E8 is very complex mathematically and takes up a lot of storage space in terms of memory on a computer, it would seem that such an expression of the complexity of nature \”could manifest\” as the \”individual mind\” using this pattern?:)\”Colour and gravity\” would remove the distinctiveness of the lines in that pattern, yet, could it be said, that a flower most appropriately is an expression of every soul?:)The intellect and the emotive states mix(entangle) and while these themselves are reflective of an inner thinking world, they are also are reflected in the world around us.Line up the world inside, overlap the schematics, integrate the emotion, and what may come about, may be quite a buzz:) Maybe an alignment of a kind that connects all three. The emotive, the intellect and the spiritual.Best,H. S. M. Coxeter, Regular Polytopes(New York: Dover, 1973) is the great classic text by a great creative force in this beautiful area of geometry (A polytope is an n-dimensional analog of a polygon or polyhedron. Chapter V of this book is entitled \’The Kaleidoscope\’….)\”