Building the Foundation: Understanding Numbers as An Infant

Yucong Duan

International Standardization Committee of Networked DIKWP for Artificial Intelligence Evaluation(DIKWP-SC)

World Artificial Consciousness CIC(WAC)

World Conference on Artificial Consciousness(WCAC)

(Email: duanyucong@hotmail.com)

Let's explore how I can build up to understanding Goldbach's Conjecture using the same approach—starting from core experiences, evolving semantics explicitly, and applying logical reasoning. This will involve developing concepts of numbers, arithmetic operations, prime numbers, and ultimately formulating the conjecture itself.

Perception of QuantityMy Observations

1. Discrete Objects:

• I notice that sometimes there are multiple objects in front of me—like blocks or toys.

• I can see and touch each object individually.

2. Comparing Quantities:

• When I have more blocks, the pile looks bigger.

• If I give away some blocks, the pile gets smaller.

Formulating the Concept of "One"

1. Identifying Single Units:

• When I hold one block, I experience a single object (dblock1d_{\text{block}_1}dblock1​​).

• Logical Proposition:One  ⟺  dsingle_object\text{One} \iff d_{\text{single\_object}}One⟺dsingle_object​

2. Understanding "One":

• "One" represents a single, discrete entity.

Developing the Concept of "Number"

1. Counting Multiple Objects:

• One block (d1d_1d1​)

• Adding another block results in a different quantity (d1+d1=d2d_1 + d_1 = d_2d1​+d1​=d2​).

• Placing blocks side by side, I notice a progression:

2. Assigning Labels to Quantities:

• Two: The quantity after adding one block to another.

• Three: Adding another block to two.

• I begin to assign labels to different quantities:

3. Abstracting Numbers:

• Each number corresponds to a specific quantity experienced directly.

• Numbers represent the count of discrete objects.

• Semantics:

Understanding Arithmetic OperationsConcept of "Addition"My Observations

1. Combining Quantities:

• When I combine one block with another, I have two blocks.

• Logical Proposition:$$\text{One}+\text{One}=\text{Two}$$

2. Consistent Results:

• This outcome is consistent regardless of the type of objects.

Formulating Addition

1. Defining Addition:

• Addition reflects the physical act of putting objects together.

• Addition is the operation of combining two quantities to form a new quantity.

• Semantics:

Concept of "Even" and "Odd" NumbersMy Observations

1. Grouping Objects:

• When I have two blocks, I can pair them evenly—no block is left without a pair.

• With three blocks, one block is left unpaired.

2. Identifying Patterns:

• Quantities that can be evenly paired: 2, 4, 6...

• Quantities that cannot be evenly paired: 1, 3, 5...

Formulating Even and Odd Numbers

1. Even Numbers:

• Directly tied to my experience of pairing objects.

• Numbers that can be divided into two equal groups without leftovers.

• Definition:

• Semantics:

2. Odd Numbers:

• Based on the observable outcome of grouping.

• Numbers that leave one object unpaired when divided into two groups.

• Definition:

• Semantics:

Exploring Prime NumbersConcept of "Divisibility"My Observations

1. Grouping Objects into Equal Sets:

• With four blocks, I can create two groups of two blocks.

• With five blocks, grouping into equal sets without leftovers is not always possible.

Understanding Divisibility

1. Defining Divisibility:

• Four is divisible by two ($4÷2=2$).

• A number is divisible by another if it can be grouped into equal sets of that number without leftovers.

• Example:

Defining Prime Numbers

1. Observing Numbers with Limited Divisibility:

• Some numbers, like two and three, can only be divided evenly by one and themselves.

• These are different from numbers like four, which have additional divisors.

2. Formulating the Concept:

• Based on the inability to divide the quantity into equal groups other than one group of the number itself or the number of one-unit groups.

• Prime Numbers are numbers greater than one that have no positive divisors other than one and themselves.

• Semantics:

Building Towards Goldbach's ConjectureExploring Sums of Prime NumbersMy Observations

1. Adding Prime Numbers:

• Eight is an even number.

• Six is an even number.

• Four is an even number.

• Two + Two = Four

• Three + Three = Six

• Three + Five = Eight

2. Patterns in Sums:

• The sum of two prime numbers can result in an even number.

Noticing a Consistent Pattern

1. Testing Even Numbers:

• $3+5=8$

• $2+6$ (But six is not prime)

• $3+3=6$

• $2+4$ (But four is not prime)

• $2+2=4$

• Four:

• Six:

• Eight:

2. Observing that Even Numbers Greater Than Two Can Be Expressed as the Sum of Two Primes:

• This seems to hold true for the even numbers I've tested.

Formulating Goldbach's ConjectureLogical Proposition

1. Conjecture Statement:

• Every even integer greater than two is the sum of two prime numbers.

2. Expressing Mathematically:

• For every even integer $n>2$, there exist prime numbers $p$ and $q$ such that:$$n=p+q$$

Evolving the Semantics

1. Grounding in Experience:

• This conjecture is not arbitrarily defined but emerges from observed patterns in numerical operations.

2. Understanding the Conjecture:

• Recognizing that the pattern may continue beyond the numbers I've tested.

• Accepting that it remains a conjecture until proven for all even integers.

Reflection on the ProcessAvoiding Subjective Definitions

• Throughout this journey, I have not relied on pre-existing definitions imposed by others.

• Each concept—numbers, addition, primes, even numbers—evolved from my direct interactions and logical reasoning.

Building Semantics Explicitly

• Numbers: Emerged from counting and recognizing quantities.

• Addition: Developed from combining physical objects.

• Primes: Identified through exploring divisibility.

• Goldbach's Conjecture: Formulated by observing patterns in sums of prime numbers resulting in even numbers.

Connecting to Real-World Semantics

• The semantics of each concept are intimately tied to my experiences and logical deductions.

• This approach ensures that my understanding is deeply rooted in reality and not abstracted away.

Implications for Mathematical UnderstandingDeveloping Advanced Concepts

• By continually building upon foundational experiences and reasoning, I can approach even complex mathematical conjectures.

• This method allows me to grasp advanced ideas without requiring subjective definitions.

Fostering Mathematical Curiosity

• Observing patterns naturally leads to questioning and formulating hypotheses.

• Encourages exploration and deeper investigation into mathematical relationships.

Conclusion

By starting from basic sensory experiences and logical reasoning, I have been able to evolve the semantics of complex mathematical concepts, culminating in the formulation of Goldbach's Conjecture. Each step was grounded in direct observations and logical deductions, avoiding subjective definitions and ensuring a strong connection to real-world experiences. This approach demonstrates how, through careful observation and reasoning, it is possible to build an understanding of even the most intricate ideas from fundamental principles.

Note: This exploration showcases how an individual, starting from basic experiences, can develop sophisticated mathematical concepts. By maintaining a focus on explicit semantic evolution, the understanding of concepts like Goldbach's Conjecture becomes accessible and grounded in reality.

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