The concepts of "space" and "extension" in mathematics


School year 2000-2001

Class: 4th year upper class
Coordinated by Pasquale DI NUNNO


This school year we have tried to clarify and examine thoroughly two key-concepts of the mathematical thinking: the concepts of "extension" and of "space".

We think that maths is a science because it is founded on an evidently or scientifically proved concept. This simple statement induces to clarify a problem: "What is the origin of the concepts of space and extension?" Is it an empirical origin?  That is, does it originate from experience? And if it originates from experience, how can an empirical concept be valid for all times and all people, namely by universally valid?

An interesting answer is given by René Descartes (1596 - 1650) in a work entitled: "Meditationes de prima philosophia" (Paris, 1641).

In the second part of "Meditationes” he asks himself:  What is the nature of this piece of wax that I have drawn from the beehive?  At first sight, its nature results from the product of some qualities: for example, the piece of wax retains the sweetness of honey, the fragrance of flowers, it is hard, cold and it produces a noise whenever I hit it. This is the objective, scientific nature of the piece of wax.

Now let's try to put this piece of wax close to fire. What happens? The fragrance of flowers fades away, the colour changes, the shape is modified, and the solid body becomes liquid. Yet, we say it is still a piece of wax! So, the nature of the piece of wax does not coincide with the product of its qualities that we have listed above!

What are we to say?

We are to say that the piece of wax is "a body" that took a particular shape at first and which appears in a different one now. To be more precise, we are to say it is "res extensa", that is "something extended".

What does it mean? The piece of wax is something that, once, it has melted, has the round shape of the pan that contains it: it could have the shape of a square or a cone as well. How many shapes? Only the ones I have seen? Or even shapes that I have not seen so far and that I can "think of"?

We should answer that the extension does not only coincide with the shapes we have seen so far, but it is, generally speaking, something that occupies a space and "to occupy a space" does not mean that the piece of wax has a specific sense shape.

So, the extension is not a simple empirical image": it is the object of "a glance of the mind" ("solius mentis inspectio").

This is a remarkably important discovery towards the enunciation that maths is a science: in fact, if the concept of extension were drawn only from experience, how could it be universally valid?

John Locke (1632 - 1704) has a different view. He deals with these problems in a work entitled "An essay concerning Human Understanding" (London, 1690).

His fundamental thesis represents the foundation of "empiricism". Which one?  The human mind is like a "white sheet of paper with no written signs, no ideas". Consequently, all the materials of our knowledge are only drawn from experience. So, if someone asked us when we started to learn, we should reply: "We started to learn when we started to make experiences". That is, the whole building of knowledge is founded on experience !

How did the concept of space take shape in us, then?  If the concept of space is drawn from experience, how did it enter our minds? Locke argues with Descartes and his followers: "In what way?" Whereas Descartes states that "body" and "extension" coincide, Locke states that they are distinct concepts. Why?

When I say that an object has a "body", I mean that object (for example a wooden cabinet) is something solid, whose parts are separable and mobile: in fact, they can be assembled in different ways.

When I say that the cabinet is something extended, I mean that between the ends of the solid and mobile parts there is a space that cannot be divided: it is a "continuum".

In other words, when I say that a cabinet is a body, I mean that it is a solid, that it resists any mechanical action like pressure or torsion. On the other hand, when I say that between the ends of cabinet doors there is a space, a surface, I do not mean that that space or surface is divisible into other surfaces!

In this extremely interesting page Locke invites us to distinguish between the "physical" reality of a body and the "geometrical" or "pure" reality of the space. And geometry does not mention this or that precise body, but pure entities, like the pentagon, the hexagon, the dodecahedron etc.

This problem is dealt with in depth by Immanuel Kant (1724-1804) in a work entitled "Critique of Pure Reason" (1780).

In the introduction Kant lists the fundamental points of his research: 
it is true that all our knowledge "starts" with experience; but, at the same time it is true that our knowledge "is not all drawn from " experience.

When I say: "The sun warms up the stone", my images of the sun and the stone and my sensation of heat are drawn from experience; but is this also true for the concept of "cause"?

And when I say: 7 + 5 = 12, is this statement true whether I am counting mushrooms or nails? What do mushrooms and nails have in common? And when I say: "The straight line is the shortest between two points", why is this true for all the straight lines that I draw?

Kant gives an answer to these questions, that are very important to enunciate that maths and physics are sciences, but stating that "space" and "time" are "pure forms", "a priori", that is prior to any particular experience of our sense knowledge.

What does it mean? Let's follow Kant in the development he traces of the theme of space.

We are accustomed to thinking that the concept of space is an empirical concept, that is, it is drawn from experience: when I was a child, I touched objects, I threw pellets, I was a little too lively; and by touching and by throwing I built the idea of space. When I say that the concept of chair or of shoe is an empirical concept, I mean that I was not born with this idea in my mind: since I have seen some shoes, I have sense representations of shoes and from these sense representations I have deduced, I have "drawn out" the universal concept of shoe. Can I say the same thing as regards the concept of space?  If the concept of space were an empirical concept, I should have a sense representation of space. But I represent to myself bodies in space and not the space in itself: consequently, if I do not have a sense representation of space, the concept of space is not drawn from experience.

So, space is not an objective quality of bodies, but if we represent "spaced" bodies, it is because our sensibility works according to the shape of space. In other words our sensibility is like an empty baking tin! If the cake dough takes on a particular shape, it is because the baking tin has given that shape to the dough. That means we ourselves "space" the world!

Here is the foundation of the universal value of geometry! This way Kant thinks he brought about a revolution similar to the one brought about by Copernicus.

Copernicus modified astronomy radically by assuming that the observer does not stand still while he regards the object under observation, but he rotates together with the object. Similarly Kant states that when we “know”, it is not the subject that conforms to the object, but the object that conforms to our forms, to our model of knowledge.

Here is a foundation of scientific nature both of maths and of physics !


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