AI Numbers
The question
I recently ask the following question to both Grok (paid account) and ChatGPT 4o (free account).
Are all numbers actually theoretical and/or approximates? 1 apple is 1 apple, but an 85g apple is less than a 100g apple. that doesn’t change the fact that it still an apple but the unit of “1 apple” is just an approximate.
The answers:
ChatGPT
You’re touching on a fascinating idea about how we quantify and define things in the world! The concept of numbers, especially when applied to the physical world, often involves both abstraction and approximation.
Counting whole objects (like “1 apple”): When you say “1 apple,” you’re using a whole number to count a distinct object. This form of counting is straightforward, and the number “1” here isn’t theoretical—it’s concrete, as you’re referring to a single, identifiable apple. But “1 apple” is an approximation in terms of its characteristics, like size, weight, or composition. No two apples are exactly the same, so while it’s “1 apple,” the physical attributes (like weight or volume) are approximations when compared to other apples.
Measuring quantities (like grams): When you measure the mass of an apple (e.g., 85g or 100g), you’re using a different type of number—one that involves measurement and precision. These measurements are never 100% exact because every measuring tool has a limit to its accuracy. For instance, when you say an apple is 85g, it might actually be 85.234g or 84.987g, but we round it to an appropriate level of precision. So, in that sense, the numbers we use for measurement are approximate.
In summary, numbers in counting (like “1 apple”) are clear, but the properties of the apple itself (such as weight) involve approximation. The world around us is often more continuous and complex than the numbers we use to describe it.
Grok
Yes, your observation touches on some deep philosophical and mathematical considerations about the nature of numbers, measurement, and classification:
Philosophical and Practical Considerations:
Ontology of Numbers: In a philosophical sense, numbers can be seen as abstract entities that exist independently of physical reality (Platonism in the philosophy of mathematics), or they could be considered as constructs of the human mind used to understand and interact with the world (nominalism or constructivism). However, when we apply numbers to physical objects, we move from the abstract to the concrete, where imperfections and variations occur.
Units and Measurement: When you say “one apple,” you’re using “one” as a counting number, which is precise in the context of discrete items. However, the attribute of “one apple” being a unit hides inherent variability:
Weight: As you noted, apples vary in weight. So, when dealing with properties like weight, we switch from counting numbers to measurements, which are inherently approximations within a certain degree of precision.
Identity vs. Properties: An apple’s identity as an apple (a discrete unit) does not necessarily encapsulate its properties like weight, size, color, etc., perfectly. Here, “one apple” works as an exact count but as an approximate when considering its attributes.
Approximation in Science and Measurement: All measurements in the physical world are approximations to some degree due to the limitations of measuring instruments, the concept of significant figures, and the inherent variability in nature.
Theoretical Nature:
Counting: When counting discrete items (like apples), numbers are exact. You can have precisely 1 apple where “1” is not an approximation but a definite quantity.
Continuous Quantities: When dealing with continuous quantities like weight, length, or time, our measurements are theoretical constructs used to approximate reality. Here, numbers often represent the best approximation within our current technological or perceptual limits.
Conclusion:
In Theory: Numbers in pure mathematics are exact. Concepts like “1” or “π” are precise.
In Application: When applied to the real world, precision meets practicality. “1 apple” as a count is exact, but as a measure of weight, volume, or any attribute, it becomes an approximation due to natural variation and the limitations of measurement.
So, while “1 apple” is indeed one apple in terms of counting, any attribute assigned to that unit (like weight) introduces variability, making the unit an approximation in that context. This reflects a broader truth in science and philosophy: the map (our models, numbers, measurements) is not the territory (the actual, variable phenomena of the world).
Summary
The winner this round is Grok.
The answer was better stated and more detail was given for a fairly difficult question.