I will keep on adding to this list.
Goal
Recreate this simple calculator app in a neural structure using all the tricks we know so far.
1. Numbers
John von Neumann’s definition of a number is that it’s a set of its predecessors.
0 = {}
1 = {0}
2 = {0, 1}
3 = {0, 1, 2}
...
Figure A
This definition transforms real numbers into unique rooms with each unique room containing other unique rooms. An empty room is 0, a room that contains the room 0 is 1, a room that contains the rooms 0 and 1 is 2 and so on.
Figure B
An experiment with a rat navigating inside three unique rooms. The concept of an empty room in Figure B and the concept of 0 in Figure A are remarkably similar.
2. Counting
The lack of a representation of sensory input in a space is essentially information.
From your video response:
As you 've said the brain represents only what it senses not what it doesn’t sense but an empty room or 0 is not something you can’t sense. It’s not something you imagine when you look around an empty room, it feels like you are sensing it. You wouldn’t need to associate an infinite number of empty location spaces to an empty room just a finite subset of all the locations your eyes saccade while you where looking will do. It’s very useful to know that the room or the cup I just looked at is empty. Why not store this information rather than store only the features and infer the emptiness somehow? @Falco Later, I can contemplate on what to add to the room or the cup.
I suspect that a better way to think of this is to focus on the concept of object representation as a collection of features. One common collection of features is the default set when no particular object is present to bind to - nothing.
This means an “empty room” is a room (set of locations) with no features associated to it. If there’s such an object that gets recognized by the absence of features it allows for a neat trick.
If all it takes is the absence of features at a location (or a set of locations) to recognize an “empty room” then every object will be recognized as an “empty room” and a unique object simultaneously through sensory input. The location space of an object is bigger than the object itself, which means it contains empty location spaces. Empty location spaces are “empty rooms” by definition. Since, all it requires is the absence of features, only a single representation of an “empty room” will be associated with each unique object. You can only infer once if features are absent or not for each object. This is different from counting how many empty location spaces are inside a location space.
Cup = Cup AND empty room
Ball = Ball AND empty room
0 = {} AND empty room
1 = {0} AND empty room = {empty room} AND empty room
2 = {0, 1} AND empty room = {empty room, empty room} AND empty room
Figure C
The “empty room” is the commonality shared between all objects and allows for a counting mechanism by association. The number 2 will be associatively recognized by any set of 2 empty rooms meaning any 2 objects not just the rooms 0 and 1 that are its predecessors in Neumann’s definition of numbers. This mechanism allows for counting and grouping of both abstract and physical objects. Also, it requires literal neural tissue that’s why it’s so difficult to count to very large numbers without devising new ways of moving in the mathematical universe.
We can use the concept of the “empty room” as the real “unit” Georg Cantor (mathematician) proposed. You can’t make a unit-based mathematical region in the brain without a window to model it from the physical space. How would you represent a “unit” ? Cantor suggested depriving objects of all of their individuating features beyond their being distinct from one another. The brain doesn’t have time to do this and it doesn’t feel like it’s doing this. When I look at a cup and a ball I know it’s two things. I don’t need to strip them down to two “units” first and then count them. The information needed in order to make this calculation must have been sensed while I’m looking at them. This is what a simultaneous sense of two empty rooms from visual sensory input can do, something not possible with units as non-sensory (abstract) objects. Also, how would you associate a unit with all abstract and physical objects? What it means to deprive them of their individuating features? The pursuit of a “unit” is a high-level, philosophical endeavor. Practically, only the concept of an empty room caters to all these needs, has an already established representation with grid cells and feels intuitive.
If you think about it an “empty room” is the “unit” because it’s what remains when you try to deprive two things of their individuating features.
The video starts from Cantor’s proposition (9:37):
3. Addition
When we are taught numbers we are also taught addition. Counting is fundamentally addition. The function of addition can be described visually as the act of an operator randomly selecting a location space as the space of the result and operate by “displacing” objects into it.
Figure D
For example, moving a “Cup” from the location space of a “Table” and a “Ball” from the location space of a “Chair” to the location space of a “Desk” has the result space of the “Desk” with two “empty rooms” associatively recognize the number 2. The result spaces of “Table” and “Chair” both associatively recognize the number 0.
"Cup" (in table) + "Ball" (in chair) = "Cup", "Ball" (in desk)
{empty room} (in table) + {empty room} (in chair) = {empty room, empty room} (in desk)
1 (in table) + 1 (in chair) = 2 (in desk)
There’s nothing permanent in this displacement. Meaning you 'll have to learn all possible displacements of all possible objects. Unless you incorporate something stable like the concept of an “empty room” and how it behaves. One way to tackle this is by starting with the obvious assumption that all numbers are re-entrant structures of the “empty room” or 0. Now, you have displacements that link all of them together, meaningfully.
4. Complex Equations
Very large numbers might as well be just more complex equations. We all have agreed to use base-10 and a series of additions.
146,525 = 1x10^5 + 4x10^4 + 6x10^3 + 5x10^2 + 2x10^1 + 5x10^0
146,525 = 100,000 + 40,000 + 6,000 + 500 + 20 + 5
We could have used 256 symbols throughout history and make a base-256 representation of the same number. It would have worked fine if we all agreed to use the same 256 symbols.
146,525 = 2x256^2 + 60x256^1 + 93x256^0
146,525 = 131,072 + 15,360 + 93
I willl randomly assign these symbols to these numbers:
2 = "!", 60 = "*", 93 = "&"
The number 146,525 in base-256 using the symbols above can be writen like this:
146,525 = !*&
This is just a way of moving in the mathematical universe and adds a layer of abstraction.
It’s obvious that algebra is done using locations and location spaces but what it means to have a negative number? Is orientation included? What about a complex function f(x) = sqrt(x) + g(x^2) as a series of other functions? Is it a sequence of displacement cells or something a lot more complicated?
If we have lived our childhood in 5000 A.D. we wouldn’t be able to get past associated counting mechanisms like what animals probably do. All later math are displacements and movements discovered by neocortexes that left their findings behind before they died. It’s not like you need to recreate all this but rather implement a very clever starting mechanism that is capable of understanding and learning all of them through exposure.
@rhyolight I’d like to know exactly how math is executed in neural structures and I feel like the representation of 0 or that of an “empty room” is the obstacle I have to overcome. I’m uninterested in philosophical inquiries.
Though people have always understood the concept of nothing or having nothing, the concept of zero is relatively new; it fully developed in India around the fifth century A.D., perhaps a couple of centuries earlier. Before then, mathematicians struggled to perform the simplest arithmetic calculations.
Zero found its way to Europe through the Moorish conquest of Spain and was further developed by Italian mathematician Fibonacci, who used it to do equations without an abacus, then the most prevalent tool for doing arithmetic.
Guys, I don’t think an abacus suits The Thousand Brains Theory of Intelligence.
