If a formula is proven by induction on n, then n is not a constant or parameter, it is a variable?
n’ = n+1 is successor.
For example of a proof by induction consider
sum 1 to n = nn’/2
for n=1, this is
a = 1*1’/2 = 1*2/2 = 1
Then for n hypothesis step
sum 1 to n = nn’/2
sum 1 to n’ = sum 1 to n + n’
=nn’/2 + n’
=nn’/2 + 2n’/2
This is n’ in the candidate formula. So it follows by induction.
==Quote from Wiki
In the identity
the variable i is a summation variable which designates in turn each of the integers 1, 2, …, n (it is also called index because its variation is over a discrete set of values) while n is a parameter (it does not vary within the formula).
Wiki thus says that n is not a variable but a parameter. But this formula is proven by induction on n, and thus n has to vary to prove the formula.
Wiki invents this strange idea that a variable has to be summed over, i.e. be a dummy variable, or it is a parameter.
Wiki proves above formula by induction, but avoids word variable.
This page has the following confused “definition”, really just a seeming definition.
Question: What’s a variable?
Answer: A variable is an object, event, idea, feeling, time period, or any other type of category you are trying to measure. There are two types of variables-independent and dependent.
Question: What’s an independent variable?
Answer: An independent variable is exactly what it sounds like. It is a variable that stands alone and isn’t changed by the other variables you are trying to measure. For example, someone’s age might be an independent variable. Other factors (such as what they eat, how much they go to school, how much television they watch) aren’t going to change a person’s age. In fact, when you are looking for some kind of relationship between variables you are trying to see if the independent variable causes some kind of change in the other variables, or dependent variables.
This is heavily into a time way of thinking and of causality and other concepts. The set of ordered pairs definition is quite different. Suppose we had the following formula
h = 2 + a * 1
where h is height and a is age in years, and the numbers are in feet.
a = h – 2
We have an inverse function, in which independent and dependent variables role are switched.
The idea that the input causes the output would be inconsistent with an inverse function. Thus the above excerpt is wrong.
This was stimulated by a comment at Keith Devlin’s blog.
re Robert B. Davis LEARNING MATHEMATICS
Part of the comment in effect stated that:
Some students don’t know what variables are after high school. Evidently that includes Wikipedia.
Notice how using the Peano Axiom framework, induction and recursion helped give us insight on the difference between variable, dummy variable, parameter and constant.
Wikipedia was out to sea and clueless in its confident declaration that n is not a variable in the formula for the sum of the first n natural numbers. In fact, the only way to prove that formula for all natural numbers n is by induction on n. Which means n varies from the hypothesis step to the induction, i.e. from n to n’ as it plays a role in the formula. Sometimes it is helpful to use a separate letter for this. The hypothesis step is then for n=k, and the induction step is for n=k’. So we do induction on k.
In this case, n varies from k to k’ as we go from hypothesis step to induction step.
By using the Peano Axiom framework, we are led to a good conceptual understanding.
When we just jump into algebra in the middle, we make up arbitrary rules on what variable and parameter mean that are wrong.
The excerpt above that is so confused also shows how the emphasis on applications is wrong. It produces wrong answers and that creates confusion that is not resolved when the students graduate high school.
The correct answer of what a variable is emerges in the context of the Peano Axioms and proof by induction. Without that structure, we get this application driven approach which leads to the wrong answer.
The emphasis on applications is misguided. This demonstrates that. The emphasis should be on what the math actually is. This refocus takes over our efforts and directs them differently. We end up unable to spend the time on applications, because the internal focus on the math itself takes up our time.
Moreover, the applications are confusing and lead to wrong ideas.
To learn the abstract idea of what a variable is, we do so by focusing on the abstract idea of what a natural number is. That leads to the Peano Axioms that include proof by induction. That tells us that n in the above formula is a variable not a parameter.
The emphasis on applications is what leads students to graduate high school with the wrong idea on abstractions like what a variable is.
Abstractions can only be understood properly inside the proper mathematical structure. Thus variable requires us to have the Peano Axioms and proofs by induction to identify what a variable is.