The New Book
Part 1: Chapter 1
Hilbert Space

# Multidimensional space

The failure of computer technology to provide an adequate conceptual framework to describe the way in which the brain functions sent scientists back to basics.

Like computers, the brain stores and processes incoming information. Like computers, it can categorize, find, select, compare and manipulate the information. Like computers, the brain can apply rules and make logical deductions. The difference being that the brain is doing this in a way that doesn't correspond to any methods or structures used by computers.

When scientists are presented with situations where the mechanics of a system are unknown, they turn to abstract models that can represent the essence of a system without the necessity of having to specify the means. As the main concern of the brain is the processing of information, the most fundamental abstract model to start with is an information space - a hypothetical space that can contain an infinite amount of information.

The trick then is to speculate on ways to format that space so that information within it can be sorted, categorized, selected, compared, combined and manipulated according to various rules and criteria. This is quite different from the conventional approach to information, where a formatted space is designed first and then information is entered into designated locations where it can be accessed by some kind of addressing system.

A suitable abstract model for visualizing an information space was first proposed at the beginning of the twentieth century by the great German mathematician, David Hilbert (1862-1943). He came up with the idea of a space with an infinite number of dimensions. It is now known as Hilbert space.

At first, this idea might appear preposterous because it doesn't seem possible that anyone can visualize a space with infinite dimensions. But, dimensions can also be called parameters, which are the elements that provide a description of an object. So, if you put every conceivable object into a space, that space could be described as having a total number of dimensions equivalent to the total number of parameters that are needed to describe all the objects it contains.

How order might be established in this space can be visualized if you take any single parameter and imagine every item with that parameter as being strung out in a line along it. For example, take the parameter "glass". All the objects in the space that contain glass would be strung out along this "glass" line.

This would apply to all parameters, so you can think of the space as being crisscrossed by an infinite number of parameter lines that can intersect with one another. Wherever a group of these lines meet at a common point the parameters and their values will be describing a unique object.

For example, where the parameters: "red"; "round"; "rubber"; "diameter 3 inches" meet in this space, the point would be describing a red rubber ball three inches in diameter.

If the parameters: "red"; "round"; "rubber" met at the parameter line "diameter 6 inches", it would be describing a red rubber ball six inches in diameter.

Where the parameter lines "round"; "rubber"; "diameter 3 inches" met a "blue" parameter line it would be describing a blue rubber ball three inches in diameter.

By suitably moving around in this parameter space, you would be able to find parameter meeting points that would be describing every possible size and color of rubber balls. By adding further parameters, this space could contain parameter meeting points that described objects of any shape, material, color or size.

The general idea is that if a Hilbert space has an infinite number of parameters (or dimensions), every possible object you could ever imagine will be present in there somewhere and they will each be located at an unique point where their descriptive parameters intersect.

Hilbert himself described this space by pointing to a glass of beer and explaining that it exists in this space at a place where the parameters describing its shape, size, materials and content intersect. He explained that if you move this object inside the space to where the content parameter is water instead of beer, you would have a glass of water. If this object is moved to a parameter where the material is pewter instead of glass, you'd have a pewter mug full of water. He further explained how by moving this object around in this parameter space you could get it to change into a beer mat, a chair, a table, etc.

The importance of this mental model is that it envisages a space where every possible thing you can ever think of is in there and every one is located at a point where their parameters meet. More importantly, not only does this space contain all the objects you know about, it also contains all the objects you don't know about. Even more bizarrely, it also contains objects that are not in existence today but could be in existence in the future.