Dear James,

I think that the metaphor map should preserve structure. Thus from the diagram we forced to consider, familiar and unfamiliar domains, as categories and the metaphor map as functor. Thus this might work for the category of manifolds. If this is the case we may ad up with a real theory of metaphors. Actually all mathematics is full of such examples.

One answer might be the one using metaphor maps, e.g.

See, attachment for the missing picture of a metaphor mapping.

Figure 1. Metaphor. The metaphor maps a familiar domain into a novel domain. It takes familiar concepts such as A and B to new concepts X and Y, and a familiar process p that leads from A to B to a new process q that takes X to Y. There are two alternate interpretations of the diagram in terms of cognitive activity. Either the brain learns to perform q on X to get to Y, q(X) = Y, perhaps using p as a model, or else the brain pulls back from the new domain to the familiar one, applies p, and then passes

back to the new domain again, M p M–1(X) = Y. The thesis presented in this paper is that the brain uses the second mechanism. In other words, the way to view the diagram is right-to-left, not left-to-right. [From K. Devlin, The useful and reliable illusion of reality in mathematics, Presented at the workshop Towards a New Epistemology of Mathematics , held at the GAP.6 Conference in Berlin, September 14-16 2006.]

Using the above map we may assert that finally «a point of view» is a metaphor mapping, in which we view the not familiar in terms of familiar.

The above metaphor mapping we may regard it as a functor from a familiar category to a not so familiar one. Algenraic topology e.g. treats topological problems in terms of group theory (homology or homotopy).

I cannot follow all your mathematical line of thoughts, sorry about that.

But in a general scientific sense: "point of view" refers to the choice of a context from which interpret experience.

To add on your quotes, remember Henry Ford's advice towards being successful in business. "If there is any one secret of success, it lies in the ability to get the other person's point of view and see things from that person's angle as well as from your own".

@Samuel Arba Mosquera: Thanks for your answer. I must note that "Metaphor Theory" is rather of linguistic origin, which have been applied to mathematics as well. As I note in my answer, e.g. one can see a function φ:X----> [0,1], as a fuzzy sets, and this is an example of a differet point of view, which is not an interpretation of some "experience"

There are many points of view in mathematics. Here are two examples:

1. View of space as a manifold of points of arbitrary dimension and defined independently of any embedding in Euclidean space. This point of view leads to the notion of a manifold in differential geometry.

2. view of geometric objects entirely encoded by the line element.

Both of these points of view come from Riemann. In terms of hyperbolic geometry and triangles from Poincare's point of view, see the attached image.

Thank you James, nice examples!

By the way haw did you manage to insert jpeg files into research gate? I try and I fail!

@Costas, use the 'Add attachment' button below with 'File' and choose the file from your HD. Done!

Dear Prof. Costas Drossos,

Many thanks for your gracious comments.

The image file for Poincare triangles is a .gif file. Try the following steps to insert an image file:

1. Put an image file in a convenient place, e.g., desktop.

2. click on Add attachment.

3. select "file" from the drop down menu.

4. use to browser to select your image file.

5. click on save.

Thank you all for the advice. I was trying to drop an image file, but that does not work. For this reason I add my pdf file for the question. Any way thank you.

In my opinion a point of view is a way in a person looks at the world or the angle from which a specific situation should be viewed; in other words a mental position from which things are viewed.

For this reason, you have different point of views depending the area and the topic you are considered. There is the point point of view of the writers, the point of view of the scientists, the point of view of the politicians, the point of view of the military, the point of view of the religious people, and so on. Not necessary all of these point of views should be the same or even similarly; in many cases could very different and even contrary. To move forward, it is important that different point of views could be merged in one unique point of view that could accommodate all views expressed about one specific topic.

@Jorge Morales Pedraza: All these are OK, but the real question is to find a kind of theory describing all possible cases. I indicated implicitly that the basis for such theory might be "Metaphor Theory". We should also note that there is a Book by Lakoff & Nunez: Where Mathematics Comes From, which try to explain all mathematics using metaphors.

Dear Costas. I think that the existence of such theory describing what a point of view is covering all areas of knowledge and all topics is something extremely difficult to find. Of course we can try to have a very general definition of this concept but in each area of knowledge we can have certain particularities that need to be considered that are difficult to be found in other areas. As you said in your remarks, one of the book try to explain all mathematics using metaphors but many of them perhaps cannot be applied in literature or even in other nature science.

Point of view can be geographical or upbringing related.

The posted pic refers. Imagine it to be a plan view of a colourful Bedoin tent. The architect can see all three colours. An Afrikaans person standing on the

East will swear that the tent is "Groen". An English person standing to the North West will swear that the tent is red!

For problem-solving, it helps to realize that you have to move around the

full circle to see how many people may describe the tent.

Then to overcome the upbringing point of view, it helps to know the vocabulary of different people around the world for the same concept.

To anticipate all possible points of view for problem-solving, you have to creatively think of all the dimensions that may affect your point-of-view.

(Did you think of colour-blindness as a possible dimension?)

That is, your anticipation has to generalize to the extent that you can cover all dimensions.

The "metaphor map" in your pdf hints is the same as work we did on "homological transfer". ("Groen"/"Green"!) Please correspond with me.

Ian,

From the pdf file, there are two spaces to consider, namely, a space X which consists of maps p: A --> B for A, B in X (familiar domain) and a space Z with maps q: C --> D for C, D in Y. Then we derive an arrow diagram of the form

A C

| ……………. |

p…………… q

| ,,,,,,,,,,,,,,,,,, |

B D

This diagram agrees with the comment in the pdf file (first interpretation), moving from familiar concepts (namely, A and B in X) to new concepts C and D in Y. Then there is a second mechanism, where one moves from moves from left to right (back to familiar ground represented by A and B in the space X). Note: I inserted rows of dots to force the down arrows to be in the position (RG is not accepted spaces between separated down arrows).

This is actually a neat idea and rings true if we consider this interpretation of point of view in the context of manifolds.

Dear James,

I think that the metaphor map should preserve structure. Thus from the diagram we forced to consider, familiar and unfamiliar domains, as categories and the metaphor map as functor. Thus this might work for the category of manifolds. If this is the case we may ad up with a real theory of metaphors. Actually all mathematics is full of such examples.

Dear Costas,

Yes, definitely, I agree. It would be interesting to pursue a theory of metaphors.

I will provide my own view point and understanding about "point of view" . For me, it is your own thinking, perception and angle of looking at objects or certain phenomenon.

Historically, the introduction of a "point o view" (or a standpoint) is the result of the bourgeoisie, specifically in the Quattrocento. On the one hand, over against the medieval worldview, which was the view of God or the church (= a view from "everywhere"), the recently emergent bourgeoisie introduced the perspective, the point of view - well, your point of view vis-à-vis mine. On the other hand, the entire space was thus conceived, particularly thanks to Brunelleschi who invented the perspective.

Thank you for your post Carlos!

A point of view may also of course be prejudiced or jaundiced.