… + … = …
no mathematics is not a natural science
mathematics is the mesaurement of consequence of movement by objects
But there is no natural science without math.
Duality = Universality
Can we comprehend a logic of triplicity?
Hi Jakob,
The cartoon is correct!
The classical Nicene Creed of mathematics reads as follows:
I believe that for every number x, x = x.
I believe that for every number x and y, if x = y then y = x.
I believe that for every number x, y, z, if x = y and y = z then x = z.
I believe for all a and b, if a is a number and a = b then b is a number.
I believe that 1 is a number.
I believe that there exists a function S such that if n is a number then S(n) is a number.
I believe that if n is a number, then S(n) is not equal to 1.
I believe that if m and n are numbers then if S(m ) = S(n) then m = n.
I believe that if K is a set, and 1 is an element of that set, and for every number n S(n) is also in that set, then all numbers are in that set.
There are a surprising number of cults:
The Algebraists who want to substitute 0 for 1. (It turns out to be OK, but I think they should do the work for themselves)
The pagan Intuitionists who want to eliminate the last article of faith.
The Formalists who want numbers to be based on ZFC set theory rather than conventional faith.
The Logicists who want numbers to be based on arbitrary signs and rules.
And there are many others.
The interesting thing is that all the cults that I am aware of, except one, derive the Nicene Creed of mathematics, (sometimes called the Peano axioms) from their pagan belief systems.
The one exception is the Intuitionists (there is a tie in to Kant, but I won’t go into that).
The advantage, that the Intuitionists have, is that fewer beliefs imply greater reliability. However, many of the modern mathematical theorems, which depend on that article of faith, would be lost if the public at large discarded that particular belief.
Thanks Ed
Mathematics is self-consistent. We don’t have to believe in the transitive property, we just have to make it so.
Hi Anthem,
Could you elaborate for me?
When you say that mathematics is self consistent, do you mean that the theorems are logical consequences of the axioms, or are you saying that the axioms are consistent with each other?
I am also not clear about the statement “We don’t have to believe in the transitive property, we just have to make it so.” I assume that you mean that when we apply the transitive property we preserve the truth of the intermediate statements. But I am not sure.
Thanks Ed
Sure. I’m saying math doesn’t require belief because it is its own system. The axioms aren’t really axioms, they’re just properties of mathematics. They only become real axioms if you wish to apply them outside the realm of pure mathematics, i.e. physics or accounting or something in the ‘real world.’
The theorems are logical consequences of the axioms, and the axioms are consistent with each other. Whether or not the axioms are logical consequences of reality might be subject to debate, but as long as you stay in the realm of pure math it doesn’t matter. Within mathematics, the axioms are always True with a big ‘T’. Change up the axioms and you can make a new system that isn’t mathematics but is also self-consistent and those new axioms are always True in that system.
So, when I say that we only have to make the transitive property and not believe in it, I was just picking a random example. I could’ve used any axiom. We picked the axioms and created a whole system that works based off them.
I compare math to language or religion or science. Each of these systems contain axioms and are self-consistent, but they might not be consistent with each other though they may all seek to describe the same thing. A Frenchman will say ‘oui’, a German ‘ja’, and an Englishman ‘yes’. They all mean the same thing, but if the Frenchman never studied another language they would assume the words the Englishman and German used used do not mean the same thing as ‘oui’, and in pure French they never will. A christian might say that the earth is 6,000 years old and a scientist might say it’s several billion years old. These two statements are incompatible with each other, but a christian can be absolutely right as long as he stays in the context of christianity and a scientist can be absolutely right as long as he stays in the context of science.
Like Nietzsche tells us, though, this does not mean all perspectives are equally valid.
That was a long answer…does it make sense?
Hi Anthem,
I appreciate your elaboration, and yes much of it makes sense.
The first four of the Peano axioms are actually logical identities which would hold whether we were considering the foundations of math or most any other subject. (I think that this helps explain much of your reaction to the transitive property).
However, the next five axioms are particularly required for arithmetic to work the way it does. As an example we don’t know and can not prove that there is not a greatest number without the use of the seventh axiom.
One cautionary note is that, while the axioms appear to be self consistent, the arithmetic that is generated by these axioms is subject to the Godel theorems. This means that we can not prove that they are self consistent.
While the concept is not conventional, I have much sympathy with the idea that the Counting numbers are simply a set with a given number of properties (corresponding to the Peano axioms). I did read somewhere that the Counting numbers could be founded on the derived properties of distribution and communitivity et al. But I am not certain how that was set up. As I see it, this type of foundation still requires the belief that such a set exists.
I agree that there can be many apparently consistent axiomatic systems, but in real life, where those systems conflict, the axioms of the failing system should be considered flawed and reevaluated.
Thanks Ed
Yes it is.
Hi Ed3, a very interesting response. Also largely beyond me, but I’d like to understand what it all means. What is the function S? Apparently I should be able to figure out the function from the definition you give, but I can’t.
Hi Jakob,
Your question is absolutely terrific! (Probably because I have asked the same question myself. OK not so terrific.)
The standard, don’t bother me kid, answer is:
S, called the successor function, is defined by adding 1 to n. More formally:
S(n) = n + 1.
If you think in terms of S(n) = n + 1 everything should make sense. (And you should probably think about this for a while).
Sometime later you might consider some of my personal questions. These are not to be confused with mathematical orthodoxy!
You might observe that addition, which is later defined in terms of the successor function, should not be used to define the successor function.
Another problem is that functions are supposed to map specified domains to specified Ranges.
The question becomes what is the domain of S? It can not simply be the number 1 because S(1) is also in the domain of S. Utimately this means that the domain of S must contain all the numbers, which seems circular because the numbers themselves are defined in terms of S.
In other words S is defined in terms of its’ domain, and its’ domain is defined in terms of S.
Any way, as you can see, I am struggling with this definition of arithmetic.
Thanks Ed
Well it’s a good thing he wrote the completeness theorem as well as the incompleteness theorem.
We can prove that complete systems are consistent, just not with that same system if it’s sufficiently strong.
But I’m no mathematician. I probably shouldn’t play with big boy words like consistent and complete. I think I was more concerned with completeness with my statement after further research.
Hi Anthem,
Playing with the big boys is not as hard as it seems. You just need a reasonably high intelligence, and the background. In my opinion you have the intelligence, but your background diverged from pure math simply because of your interest in engineering.
One of my favorite things about you is that you actually do the research once you get into a subject.
Thanks Ed
On the contrary I am happy having asked not a stupid question but a sensible one. Or maybe they’re the same.
So it defines the mechanism of counting?
Indeed, I had to. I can absolutely not think in the letters of the alphabet in this way. It takes time to see beyond them and see the simplicity.
What kind of questions do you ask?
I think I see; the successor is automatic, a mechanism, addition is a collection of acts or relations which make use of the successor function.
It is what it is and that’s all it aspires to. I guess it’s pretty natural.
Hi Jakob,
Yes, S(n) does define counting.
According to the Peano axioms, all we really know is that 1 exists. But S(1) generates the next number which turns out to be 2. Now S(2) generates the next number which turns out to be 3 and so on.
The whole numbers starting with 1 are actually called the Counting numbers.
Formally, starting with 1, the number n is defined to be the (n - 1)th successor to 1. So the number 2 is the first successor to 1, and 3 is the second successor to 1 et cetera.
Addition is later defined on two numbers n and m, generally written n + m, as the mth successor to n.
Therefore n + 1is the first successor to n or S(n) = n + 1.
These axioms provide a framework that works for most people. However they are not satisfactory for many others as the large number of variations attest.
More Personal Observations.
I personally am uncomfortable assuming the existence of a function S whose domain is defined by that function. (You should be clear about the fact that the Counting numbers have not been defined at this point).
Another issue is whether or not S is unique. This is a common question in mathematics, particularly in differential equations, but it pertains generally. I belive the answer is No!
For example let S1 be an initial function that satisfies the Peano axioms. If we define S2 such that S2(n) = n + 2, then we could define 1* = 1, 2* = S2(1), and 3* = S2(2) et cetera. (These correspond to the odd numbers in the first system). In general n* would be the (n* - 1*)th successor to 1*. If this conjecture is correct, and I think it is, then there are multiple Counting number systems that behave the way we want.
I have been trying to find some supporting evidence, but if you do a Google search for “successor function unique”, you will find the anomaly that the descriptive text indicates that S is not unique and yet in the body of the first few articles the evidence is missing.
Thanks for your responses.
Ed