by Steven Gibson January 2006

**Abstract**

This paper proposes a model to explain the usefulness of the number system. The number line and basic number theorems are used daily by people to make predictions and explanations of real world events. People commonly believe arithmetic works and this paper offers one systematic model to explain it. The proposal is that a 1-dimensional cellular automation can be used for modeling the correspondence between the real world and features of the arithmetic number line. I postulate that any other universal computing tool could also be used for the model. So reversible Turing machines and lambda calculus could also be used for this modeling.

Numbers are posited to be human artifacts that model processes like measurable objects in space-time. The paper details the differences between real world space-time energy-matter and the human artifact of the number line. Then we offer a model that bridges the two. It is postulated that number features are expressed with symbols and a finite sequence of instructions.

While the domain covered by this model is limited to positive integers of the natural number sequence and basic arithmetic, some unexpected implications about number theory will be stated for future exploration.

**Beginning Assumptions**

The use of numbers has been a tool that has raised humankind to the heights of technology that it has achieved.

The proposal is that a 1-dimensional cellular automation can be used for modeling the correspondence between the real world and features of the arithmetic number line.

Our belief about space-time postulates is that the common sense notion of numbers and counting is only valid because of created structures. Selecting a number corresponds to slicing the system over an arbitrary area.

Other assumptions are that discrete space-time and matter do not exist in the real world, but humans approximate the appearance of discrete units, and it works.

Time is a human representation of the change states of space-time.

Here it is postulated that numbers do not exist, but we can work with abstract objects that work. We can create units that are usable in the real world. We are able to create a practical use even though there is tension between discrete and continuous points of view.

Numbers are a creation of our minds, while space and time have an independent reality that must be mapped out experimentally. I here assume arithmetic is the work of mind and constructive methods can produce experimental solutions. The ontological framework used here is that all entities are parts or facets of one unified continuum

**Energy/Matter**

The postulate in this paper is that events and objects in the world are representing states of matter and energy. All matter and energy is a view of one whole. Energy is a human abstraction of states of objects or events that exist in the world. Matter and mass are assumed to be other states of the same entities that are measurable as states of energy.

Space-time is the arena on which all physical events take place. We observe that space-time displays discrete and continuous forms.

The real world in large scale and small scale does not agree with intuitive human ideas. Our model for real world events is not grounded in our human senses, but must be developed and tested by instruments of observation and measurement. These instruments are the result of constructs based on existing or proposed theory. We postulate that spatial state change produces observable objects in space. By observing and measuring events and objects we are extracting usable order out of the complexity of space-time.

Energy/matter seems to be distinct because:

1. Our visible observation equipment detects differentiated objects

2.one unit of energy/matter can interpermeate other energy/matter\

Number-line/Arithmetic

Numbers are postulated to be human constructs. Simple rules lead to the number system. Computation proceeds by making changes in a coherent way. Physical systems are countable based on measurable information. We create an individual number by selecting a portion of energy/matter we see as standing out of the energy/matter continuum. So a number is an abstract symbolic object we create based on real world events/objects.

The use of numbers is a result of human cognitive requirements and language/symbol tools. We postulate each number being a discrete repeat of any other one. The labeling of a second “2” is an arbitrary human artifact. A one implies the lack of no number, meaning not-one or zero. For us to identify one we must be able to separate from the not-one.

Some postulates about numbers:

they are constructs of human thought

The number line as such is a model of representing and manipulating the symbolic number

some of the qualities that are expressed in number theory result from how we conceptualize numbers other than 1

the elementary number theorems seem to be based in our experiences with the real world around us

Numbers work because we can use language constructs to describe the world. We can record these constructs and take action based on them and cause a predicted effect in the real world. Objects, and events exist in the real world. Which can be described with language and used. Our interpretations of the world and actions in the world depend upon the sense tools we can use to view the world. and the action tools we can use to act in the world. So these representations entail concepts determined by real actions. Mathematical terms, for instance the concept of number, are explicitly defined and are abstract insofar as they are determined within the logical structure of a language. They also refer, to concrete objects and actions. So numbers, have an abstract structure on the one hand, on the other a connection to the arithmetical activities that work in the real world.

The ontology of this approach is that the assertions of number theory have no objective truth values, independent of the conventions, languages, and minds of the mathematicians.

**Cellular Automation/Bridge Model**

The postulated model here is that we can use cellular automation to model the correspondence between the real world and features of the arithmetic number line.

Cellular Automation are discretely defined but exhibit continuous dynamics so are useful to address the discrete and continuous.

A cellular automation can model usable calculation algorithms. These models are idealizations that capture some features and ignore others. Mathematical tools are embedded in structures in natural language. The elements for this bridging model include rules, procedures, transformation and rewriting. We need to model a arithmetic system with algorithmic methods. The algorithm contains numbers and follows a set of steps. Prediction of adheres to a program's run. The program plus its initial condition moves to the end task.

The reason CA works as a bridge model is because:

A. it gives the correct results for basic arithmetic

B. it models actual real processes in the energy/matter continuum's

i) algorithmic steps forming a sequence

ii)individual units that change through the action of the steps

A step in a change rule is a string of simple elements. Change rules apply rules repeatedly and model to energy, space and time.

This model uses a 1 dimensional first order cellular automata with the following definition:

This 1D binary CA is defined as where

Z can be finite or infinite

S = {0,1} is a set of two values

N = { − 1,0,1} is the neighborhood of size k = 3 with symmetric radius k0 = 1

f is a transition function rule

B = {b − 1,b1} is the boundary

1 -> 000

2 -> 001

3 -> 010

4 -> 011

5 -> 100

6 -> 101

7 -> 110

8 -> 111

This CA is reversible. This CA function mapping is bijective.

And from H. Morita we find invertible Turing Machines can be simulated by invertible cellular automata.

**Conclusion**

The conclusion is that the numbers, number-line and basic arithmetic are not Platonic objects but are human created tools.

It is shown that the usefulness of numbers can be explained by models of human language and models that explain the usefulness of the space-time postulate.

The number-line is stated to be a tool created by humans to be used in algorithms. Humans posit real world events that behave like numbers. Individual objects are posited to be useful within the present space-time theories. A model that produces the traits of the real world, and of basic numbers and arithmetic is a simple universal calculating system like cellular automation. Cellular automation has traits that model the real world and can emulate and reproduce arithmetic methods.

The number-line models energy in real space with two restrictions

1.the numbers are restricted to 2 directions

2.the cases are limited to full units (integers)

The number-line requires two aspects

A. a symbol set, and

B. a program that uses the system set

Programs require at least one function definition and can have a transformation rule. A program is defined to be a finite sequence of instructions.

A computable number is a number for which there is some program to compute it. Turing machines or cellular automations can model such computation

A model for 2 dimensional arithmetic can also be modeled with similar CA. Basic forms of behavior are modeled by cellular automation, Turing machines or lambda calculus. Arithmetic laws are validated in ways that other mathematical postulates are verified.

**Future Steps**

The linear number system may be a special case of a number system model that could be built that is 3 dimensional. This could lead to models that demonstrate some constants like pi and cosine and perhaps explain irrationals like square root(2).

Numbers arise because of the trait that processes have of being able to be sliced through a portion of time and change can be observed or likeness can be seen.

Numbers can be thought of as a abstract model that intrudes into a small part of the complete space/time of a process with measurable abstract objects.

There is perhaps work to be done using order theory to move from individual objects to the number-line. We need a systematic approach to object identification. Such a system would be foundational to arithmetic. To have usable objects we must be able to individuate entities out from the continuity and to identify these individual entities over time

**References**

1.Rudy Rucker, Mind Tools: The Five Levels of Mathematical Reality,

1987

2.S. Wolfram. A New Kind of Science, Wolfram Media, 2002.

3. George Lakoff, Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being, August 2001

4. Jurgen Habermas, The Theory of Communicative Action: Reason and the Rationalization of Society, Social Science, 1985

5.Wilfried Sieg, (following Turing and Gandy),Calculations by Man and Machine for an abstract setting for computation

6.Konrad Zuse, Calculating Space, 1969

7.J. H. Conway, On numbers and games, second edition, A. K. Peters, 2001.

8. Edward Fredkin Did the Universe Just Happen Altantic Monthly April 1988,

9. Stewart Shapiro, Modality and ontology - philosophy of mathematics,

Mind, July, 1993

10. Peter Lynd, Time and Classical and Quantum Mechanics: Indeterminacy vs. Discontinuity, 2003

Presented at Wolfram Science Conference June 2006 - Washington DC

http://www.wolframscience.com/conferenc

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