Are the Laws of Arithmetic Empirically Derivable?

[kant]

One stone plus 5 stones equals 6 stones. Is that a unique property of our universe, or it is conceivable that in a different universe with a different set of physical laws, arithmetic is impossible?

 

[-Job-]

The concept of a "unit" is necessary in addition. Without units there isn't anything to add. Humans experience a "quantized" Universe where everything that is perceived is automatically grouped into units. From our perspective of the Universe addition arises as a basic relationship.
I can identify two units of rocks on the left and four units on the right or 6 in front of me, all by the same process. This leads to a relationship between 2, 4 and 6.
Addition is more of an observation than a property, and not independent of a mind.

 

[Doctordick]

One stone plus 5 stones equals 6 stones. Is that a unique property of our universe, or it is conceivable that in a different universe with a different set of physical laws, arithmetic is impossible?

You make two very different errors in your question. First "one stone plus 5 stones equals 6 stones", is a statement extracted from a mathematical construct called arithmetic. Arithmetic is a collection of statements and deductions presumed to be an internally self consistent system (many people have thought about it for many years and they have come to agree that the system does not invalidate itself via contradiction). The possibility that someone can conceive of an internally self consistent set of definitions and actually set one up seems relatively clear.

That brings us to your question: "is it conceivable that ... ". What is and is not conceivable certainly cannot be judged via your (or anyone else's) ability to conceive of it. Speaking of conceiving of an internally self consistent set of definitions, you might consider your ability to conceive of a wholly different (and internally consistent) interpretation of the English language. The volume of information which needs be reinterpreted to accomplish that result is certainly a trivial fraction of the information represented by the idea "the universe". So it pretty well follows that our ability to conceive of things is clearly somewhat limited.

I can come up with no evidence that your interpretation of the meanings of the terms of the English language are the same as mine. All I really "know" is that my interpretation seems to make sense of the language; perhaps yours is totally different and still makes sense as a whole. Show me wrong if you can. Or better yet, show me where you would start with such a demonstration.

Have fun -- Dick

 

[kant]

You make two very different errors in your question. First "one stone plus 5 stones equals 6 stones", is a statement extracted from a mathematical construct called arithmetic. Arithmetic is a collection of statements and deductions presumed to be an internally self consistent system (many people have thought about it for many years and they have come to agree that the system does not invalidate itself via contradiction). The possibility that someone can conceive of an internally self consistent set of definitions and actually set one up seems relatively clear.

hmm... can you tell me how you show the arithmetic statement "arithmetic is consistent" to be true?


i guess arithmetic is based more or less on the idea of discrete units, quentities. I can centainty imagine a null universe with no space-time. In such case, it seen there are no discrete things, so might we abstract that arithmetic is impossible?

That brings us to your question: "is it conceivable that ... ". What is and is not conceivable certainly cannot be judged via your (or anyone else's) ability to conceive of it.Speaking of conceiving of an internally self consistent set of definitions, you might consider your ability to conceive of a wholly different (and internally consistent) interpretation of the English language. The volume of information which needs be reinterpreted to accomplish that result is certainly a trivial fraction of the information represented by the idea "the universe". So it pretty well follows that our ability to conceive of things is clearly somewhat limited

i can reinterpret english using french or chinese, or change the name "english" to "chinese" etc... You are make a underlying assumption that are ability to conceive something is based on our ability to reinterpreted some thing. I think this is a mistake

 

[Doctordick]

hmm... can you tell me how you show the arithmetic statement "arithmetic is consistent" to be true?

I'll leave that to the mathematicians.

i can reinterpret english using french or chinese, or change the name "english" to "chinese" etc... You are make a underlying assumption that are ability to conceive something is based on our ability to reinterpreted some thing.

No, I was just pointing something out to you that you find inconceivable. Relabeling the concepts in a different symbolic system (French, Chinese, etc.) has nothing to do with the validity of the concepts. What I was pointing out was that you can not comprehend the collection of concepts you attach to those words being the wrong collection of concepts (that possibility is thus something you cannot conceive of).

Have fun -- Dick

 

[moving finger]

One stone plus 5 stones equals 6 stones. Is that a unique property of our universe, or it is conceivable that in a different universe with a different set of physical laws, arithmetic is impossible?

Whether the proposition is true or not depends on one's premises (the definitions of "one", "5", "6", "plus" and "equals"). If you can provide definitions of these terms, then we can judge the truth of the proposition.

(it has nothing to do with physical laws, only with one's premises and the laws of logic)

Best Regards

 

[octelcogopod]

So basically the physical laws are different from the logical laws.

The physics might change but the logic remains the same?

So for instance in another universe we have 5 stones and 1 stone, if we were to travel to this universe, we would see 5 stones and 1 stones as equal to 6 stones, from our point of view, but others who were born in that other universe might have other logic for it?

Or is that even possible?
It seems to me that if arithmetic was to be impossible in another universe, we couldn't use spatial or quantified time/space.
If you have a universe where there is a quantity of something, then that quantity will always be constant, even if the laws change rapidly, even for just a split second.

So basically, no, 5 stones and 1 stone will always be 6 stones (If the other universe has a quantity of stones)

 

[matt grime]

Here is an example of something, though what is debatable.

Suppose we have two races who both have a concept of quantity, that is they can look at a pile of things, like stones, and produce a symbol. Suppose that someone constructs a dictionary relating the two labels directly. So, if there is a pile of stones and we ask race A what it is, and tehy say X(A) we can look in the dictionary and find out that race B would say X(B). Now, here is the question - is the concept of addition somehow intrinsic to the dictionaries? More explicitly, if both races have a concept of additions and if X(A) + Y(A)=Z(A), does it follow that X(B) @ Y(B) = Z(B) where + and @ are the rules A and B use for their concept of addition? The answer is no. More mathematically there are more then two additive structures defined on the set of integers, eg x@y = x+y+1 with + the normal addition. Thus 1@1=3 in this system. To understand what the system is saying really (and why it is in some sense natural) you should think of the numbers as being the spaces between objects, not the objects themselves. For example, 1 can be seen by holding up two fingers and noting there is '1' space, put one more space next to it and you have 4 fingers, so three spaces.

 

[oldman]

The concept of a "unit" is necessary in addition. Without units there isn't anything to add. Humans experience a "quantized" Universe where everything that is perceived is automatically grouped into units. From our perspective of the Universe addition arises as a basic relationship.
I can identify two units of rocks on the left and four units on the right or 6 in front of me, all by the same process. This leads to a relationship between 2, 4 and 6.
Addition is more of an observation than a property, and not independent of a mind.

I fully agree with what you so straightforwardly say. A concept like "2", or "4", together with the concept of addition and indeed the rest of mathematics, are constructs of the human perspective which are not independent of a mind. In my view such entities have no physical existence. Useful and fun for us, though!

I guess that other minds in other universes could devise similar constructs if they wished, in answer to Kant's original post. Sadly, we will never know.

I made similar points in a recent thread about The Platonic World, but got sidetracked into matters stringy.

 

[moving finger]

I fully agree with what you so straightforwardly say. A concept like "2", or "4", together with the concept of addition and indeed the rest of mathematics, are constructs of the human perspective which are not independent of a mind. In my view such entities have no physical existence. Useful and fun for us, though!

I don't agree that these are simply "constructs of the human perspective". The fact that 2 + 4 = 6 is an analytic truth (a truth by definition) - the same way that "all bachelors are unmarried men" is an analytic truth. Given the definitions of 2, 4, 6, + and =, it logically follows that 2 + 4 = 6 (in all logically possible worlds).

Best Regards

 

[oldman]

I don't agree that these are simply "constructs of the human perspective". The fact that 2 + 4 = 6 is an analytic truth (a truth by definition) - the same way that "all bachelors are unmarried men" is an analytic truth. Given the definitions of 2, 4, 6, + and =, it logically follows that 2 + 4 = 6 (in all logically possible worlds).

Best Regards

And what is "an analytic truth (a truth by definition)" , if not a "construct of the human perspective" (a.k.a. words, or put more crudely; hot air)?

But I don't want to denigrate the usefulness to us of logical hot air. This substance has bouyed the balloon that is civilization and promoted the efflorescence of humanity very well, thank you.

 

[moving finger]

And what is "an analytic truth (a truth by definition)" , if not a "construct of the human perspective" (a.k.a. words, or put more crudely; hot air)?

Are humans the only agents capable of producing hot air?
Are all words (and logic) necessarily hot air?

imho, logic exists independently of the existence of human agents

Best Regards

 

[oldman]

To answer the questions Moving Finger raised in post # 12:

1. No. We're beginning to realize that we aren't the only folk who chatter intelligibly to other members of our species. Our fellow primates and birds also generate such hot air.

2. No. Words can exist also on paper and on computer screens. One might call such manifestations "virtual" hot air.

3. You are not alone in your opinion. Sir Roger Penrose seems to share it. But I have yet to be argued into agreement on this point.

 

[moving finger]

3. You are not alone in your opinion. Sir Roger Penrose seems to share it. But I have yet to be argued into agreement on this point.

Darn it. And I thought I disagreed with Penrose on most things.

Best Regards