“I think you are confusing knowing that a system will perform arithmetic, with the system actually performing arithmetic. The latter does happen sometimes, despite all fallible assumptions.”
I think you didn’t understand my argumentation; when you say that a physical system does perform arithmetic, then your theory of arithmetic is wrong as soon as you have a contradicting result.Therefore the system is not allowed to perform arithmetic sometimes, but it is required to do it always!
Let’s consider this: I find a machine I don’t know anything about it. I soon find out it has two input dials and what looks like an output register.
By experimenting with the inputs and noting the outputs I found out that the inputs are presumably decimal numbers and the output looks like the arithmetic sum of these numbers.
I say: ‘Oh, looks like it adds two numbers.’; now I’m using this machine many many times, until one day where the result isn’t the arithmetic sum (let’s assume there is no overflow).
‘Bugger, seems this machine is broken...’
Now, the 1M $ question: ‘At which point did the machine got broken?‘. Did it get broken exactly at the point when it printed the wrong result? But what if the inner workings had the defect way before? And it only prints ‘wrong’ results for specific inputs?
You clearly don’t want to question your theory of arithmetic (because your theory doesn’t have any contradictions). But let’s assume that the creator of this machine didn’t want it to perform addition, but he wanted it to calculate the Foobar-value of two ‘numbers’. The Foobar-value looks like addition for a majority of values, but for some combinations it’s something completely different.
Of course you can examine the inner workings of the machine; but if you don’t know the intentions of the engineer, you perhaps find the special part that is responsible for the ‘wrong’ results. You can ‘fix’ the machine by replacing that part (assuming it’s an engineering mistake), or you assume it doesn’t calculate the sum of two numbers and try to find out what this Foobar-value might be for.
But we can avoid this problem by knowing that theoretical devices work on different domains than physical devices. And this is, what technology/engineering is all about: Find a mapping between both, that is reasonable under real-world constraints.
@Elizier
“I think you are confusing knowing that a system will perform arithmetic, with the system actually performing arithmetic. The latter does happen sometimes, despite all fallible assumptions.”
I think you didn’t understand my argumentation; when you say that a physical system does perform arithmetic, then your theory of arithmetic is wrong as soon as you have a contradicting result.Therefore the system is not allowed to perform arithmetic sometimes, but it is required to do it always!
Let’s consider this: I find a machine I don’t know anything about it. I soon find out it has two input dials and what looks like an output register.
By experimenting with the inputs and noting the outputs I found out that the inputs are presumably decimal numbers and the output looks like the arithmetic sum of these numbers.
I say: ‘Oh, looks like it adds two numbers.’; now I’m using this machine many many times, until one day where the result isn’t the arithmetic sum (let’s assume there is no overflow).
‘Bugger, seems this machine is broken...’
Now, the 1M $ question: ‘At which point did the machine got broken?‘. Did it get broken exactly at the point when it printed the wrong result? But what if the inner workings had the defect way before? And it only prints ‘wrong’ results for specific inputs?
You clearly don’t want to question your theory of arithmetic (because your theory doesn’t have any contradictions). But let’s assume that the creator of this machine didn’t want it to perform addition, but he wanted it to calculate the Foobar-value of two ‘numbers’. The Foobar-value looks like addition for a majority of values, but for some combinations it’s something completely different.
Of course you can examine the inner workings of the machine; but if you don’t know the intentions of the engineer, you perhaps find the special part that is responsible for the ‘wrong’ results. You can ‘fix’ the machine by replacing that part (assuming it’s an engineering mistake), or you assume it doesn’t calculate the sum of two numbers and try to find out what this Foobar-value might be for.
But we can avoid this problem by knowing that theoretical devices work on different domains than physical devices. And this is, what technology/engineering is all about: Find a mapping between both, that is reasonable under real-world constraints.