# The Nature of Mathematics

I would like to explore the relationships of mathematics to the empirical world and logic. I assume that any one contributing to this discussion has read about the various foundations of mathematics and has some background in logic.

Suggested minimum background:

The axiomatic Foundations of Arithmetic:
en.wikipedia.org/wiki/Foundations_of_mathematics

Before we can explore the relationship of mathematics to anything else we should talk about the various foundations.

My first question for discussion is:

Which if any of the foundations of mathematics, or groups of foundations do you prefer and why?

Related Questions:

Do you believe that all provable statements in Mathematics follow logically from a set of axioms?

What is the role of a definition in Mathematics?

Finally, while this touches on logic, what do you think about the constructivists approach?
Can you touch on the role of the excluded middle and the lack of transcendentals, notably pi and e?

We will restrict our discussion of the empirical world to one which contains objects that can be quantified or measured in some manner. It is permissible to expand on or modify this definition.

Regarding the relationship of the empirical world to Mathematics:

Describe your understanding of this relationship. This should be done bearing in mind that relationships can be very general.

Can you differentiate your responses with respect to the Ideal or abstracted foundations versus the various axiomatic foundations?

Regarding the relationship of Mathematics to Logic:

What is logic? Personally I like Faust’s post on logic.

But there are a few variations. Please expand if you are so inclined.

For a time, it was conjectured that Mathematics arose from logic. In the literature this conjecture is generally regarded as discredited. Is there more to be gleaned from this subject?

What do you see as the role of Model theory (constraining the objects and relationships in logical propositions to specified sets, relations, and functions) with respect to logical propositions?

Can you give examples of where this constraint provides a more precise proposition than a conventional logical proposition? What are the drawbacks?

Thanks Ed

Some thoughts on the subject:

This is not simple.

When I do math I primarily think in terms of sets.

However, as ZFC is formulated, I do not like the axiom of existence due to its‘ lack of reference. The authors of the Wiki article on ZFC, it appears to me, also avoid that axiom, by restating it in terms of other ZFC axioms. (One of the problems of ZFC is that it has too many axioms).

On the other hand the abstracted math that we learn in school has better philosophic qualities. Since 2 is the abstraction of two apples, two oranges, two bananas…., it appears to be “stickier” for lack of a better word to the empirical world.

Just a small note about two apples:

The two in two apples is not the same as the 2 that we abstracted from two apples two oranges and two bananas….

The twos are all adjectives and are part of the empirical world that we experience.

The abstracted 2 is a noun and a mental object that we discover.

But many times two apples and three more apples is the same as (2 + 3) intuitively unabstracted apples or 5 intuitively unabstracted apples; and we should further explore the relationship between Mathematical objects and the empirical world.

Some of our more perceptive philosophers will notice this difference but I am afraid that many elementary math schoolbooks will skip over this distinction.

Platonic Idealism can be rejected due to Occam’s razor.

On axiomatic foundations:

I prefer Russell’s Principia Mathematica (the logicism approach) to the Peano axioms because I think that Peano axioms are circular.

I do not like Brouwer’s Intuitionism because of its’ subjective nature and I strongly disapprove of its’ rejection of the excluded middle. Wittgenstein’s foundation is better because it lacks Brouwer’s subjective nature but it also suffers from a rejection of the excluded middle. (There is an interesting equivalency between Wittgenstien’s foundation and a primitive programming language called lambda calculus; but I think that a programming language is not a substitute for a robust mathematical foundation).

Concluding Remarks on the various foundations:

So now we have two classifications of mathematical foundations. One is axiomatic and generally considered to be created and the other is abstracted and generally considered to be discovered.

There is a nice article on these two foundations and how they relate the mathematical world to the empirical world on page 80 of the August edition of “Scientific American”. But I do not consider the article to be rigorous.

Related Questions:

Do you believe that all provable statements in Mathematics follow logically from a set of axioms?

What is the role of a definition in Mathematics?

I do not believe that mathematics follows logically from axioms. In fact as was mentioned in the body of the OP I believe that this issue is generally regarded as settled. The main example given is the two non Euclidean geometries resulting from the variations of the 5th Euclidean postulate.

However, I do not believe that one needs to resort to axiom variations. Simply by adding definitions in a theorem or the body of a theorem, I think that one destroys any logical path to a theorem.

One of the best examples of a created mathematics that I can give is in a “Big Bang Theory” episode where Sheldon invents a three-person chess game where he invents two extra pieces and gives the rules for their movement. As usual the physicist (Leonard) and the normal person (Penny) leave the room in disgust.

This is something that one generally does not learn from a mathematics textbook.

Feedback is welcome and I will try to complete with my remaining thoughts over time.

Ed

The constructivists:

Brouwer: numbers are a personal recognition of the passage of time. This is finite for every person and so numbers must take an unspecified upper bound. (We don’t know what that is).

Wittgenstein: Numbers are of the form [0, x, Op(x)] and are generated by the operator Op. This is a rule driven definition but the operator Op is only allowed to operate over a finite number of inputs.

To contrast the constructivists view with the conventional view I will give a simple classical proof that the Counting numbers are infinite.

Assume not:

Then there would be a largest number, call it N, such that no other number is larger than N.

Now add 1 to that number. We can do this because it is a property of every major axiomatic or abstracted foundation.

Then N + 1 is a number.

But since N + 1 is greater than N, N can not be the largest number.

This is a contradiction and therefore our assumption that the Counting numbers is finite must be false.

We conclude that the Counting numbers are infinite.

This presents a huge problem for the constructivists who don’t want to deal with infinite sets. In order to avoid these problems, Brouwer, Wittgenstein (who studied with Brouwer), and constructivists in general, disallowed the law of the excluded middle.

By disallowing the excluded middle they discard both the following conventional laws:

If A implies B and NOT A implies B, then A is False
NOT(NOT A) = A

If we can not assume that NOT(NOT(A)) = A, then we can not produce a proof by contradiction. This represents a major blow to conventional mathematics. It is hard to estimate the damage, but I would estimate from 1/3 to 2/3 of all mathematical proofs would be lost. Additionally, restricting ourselves to only finite searches ultimately means that we can not include the transcendental numbers pi or e as Real numbers. These are serious issues to any working mathematician.

Can the loss of the law of the excluded middle be justified in an n valued logic?

We might think that it is possible to look at multivalued logic such as a three valued logic where the truth tables take on the values of True, False and Unknown. The logic has some practical applications. But I have tried to work out the truth tables, assuming the excluded middle, for NOT(NOT(A)) and failed to match the conventional Kleene or Priest results.

Very interestingly, in 1932 Godel proved that Intuitionist logic is not a finitistic valued logic!!! A reference to this can be found at:

en.wikipedia.org/wiki/Many-valued_logic

I believe that this is a very serious blow to the mathematical foundations of both Brouwer, and Wittgenstein; and the agenda of the constructivists in general. Basically if you want to deal with only finite sets, you must use a non finite valued logic. SORT OF DEFEATS THE PURPOSE!

A Personal Observation:
Despite the fact that we can not apparently rid ourselves of the excluded middle without some serious consequences, there appears to me to be a strong preference for a constructivist view. From my small sampling, it appears to me that the Wiki authors will give a constructive proof over a much simpler proof by contradiction; and the author of the English translation of the Godel theorems trumpets Godel’s supposed constructive solution in the introduction.

End of Personal Observation

Thanks Ed

Excellent posts, Ed. I wish I were up to speed on the philosophy of maths enough to contribute meaningfully!

A question arises from the points above:
Wittgenstein: Numbers are of the form [0, x, Op(x)] and are generated by the operator Op. This is a rule driven definition but the operator Op is only allowed to operate over a finite number of inputs.
vs
Now add 1 to that number. We can do this because it is a property of every major axiomatic or abstracted foundation.

There seems to be a conflict there; either the operator is restricted, or it’s axiomatically not. No?

Hi Only_Humean,

Thanks for your post. I think it is a good one.

I suspect that I did not phrase the following comment appropriately:

“Op is only allowed to operate over a finite number of inputs.”

Additionally it might be helpful to mention that Op(x) = x + 1, which is virtually synonymous to the successor function in the Peano axioms.

The best analogy that I can give is the following computer algorithm:

Dim n As Integer
Let n = 0
Label A
n = n + 1
Print n ’ this is our new number
Goto Label A
End

While this program loops, n + 1 is always a new number, but we will never generate an infinite number of numbers or for that matter an infinite number of loops.

It should also be clear from this analogy that Wittgenstein’s foundation is flawed. i.e. Unless you already know what the integers are, (The Dim n statement) you can not add 1.

Thanks Ed

Correction:

In the post on the constructivists I wrote:

I should have said:

But I have tried to work out the truth tables, assuming the excluded middle [is not the case], for NOT(NOT(A))

Thanks Ed

Hi to All,

I wish to continue my responses to my original OP. This post is very conventional and really just a setup for the post to follow.

The empirical world:

We define the empirical world as the world we experience.

However, abstracted arithmetic can not universally be derived from the empirical world. Clouds, heaps of salt, and subatomic particles (virtual particles popping in and out of existence) do not act in a manner that can be clearly arithemitized. We will need to constrain our interest to an empirical world that can be stably quantitized. i.e. These objects need to have a property that we can talk about like: one banana, two apples, three oranges et cetera.

Furthermore, I personally prefer to deal with a harder reality, one that others can generally agree upon. Basically, this discussion should deal with a world that philosophically is considered a part of public knowledge.

Similarly, mathematical objects and relationships should be mental objects and relationships that others can generally understand at least on an elementary level such as arithmetic or geometry.

NOTE: If you disagree with a public knowledge standard, I would encourage you to speak out as it is clear that there are shortcomings.

From hereon I will restrict the conversation to this type of constrained empirical and mathematical world.

The question I wish to address is:

“What is the relationship between mathematics and this constrained empirical world”?

How does the empirical world relate to mathematics?

Clearly, the empirical world can generate the abstracted arithmetic that we learn in school. This includes the Counting numbers, addition on those numbers, and an ordering system. (We can say that m is less than n if and only if there is a Counting number r such that m + r = n). But are there other aspects of the empirical world that can impact our view of mathematics?

Consider a convenience store. In many cases they have apples for sale and we will assume that this is the case. If you have shopping cart you can start loading those apples into your cart. Each time an apple is loaded into the cart you can mentally count the apples with our newly minted abstracted Counting numbers. The total number of apples in the store, we assume without loss of generality (otherwise we get more carts), can be counted.

The total number of apples in the cart(s) is defined as the cardinality of the apples in that store.

There are a number of different ways to determine cardinality, the most common of which is much more laborious, but, assuming that the number is finite, they will all arrive at this same number.

Since the store and the empirical object were arbitrary, we will assume that this is true for any of our constrained empirical objects.

So now we have empirical objects that define our abstracted mathematics; and they define the cardinality of any finite collection of empirical objects. I believe that there are numerous other examples of how the empirical world affects mathematics, but the Counting numbers, addition, ordering, and cardinality are the major concepts that I will need to use in showing how mathematics affects the empirical world.

Thanks Ed

Hey Ed! First time back in a while and I just caught your post. Excellent topics as always. I don’t have time to give a comprehensive answer of my perspective, but I can give several answers and a good reference in one:

Category Theory.

http://en.wikipedia.org/wiki/Category_theory

Category Theory is a beautiful component of the framework of “logic”. If you’re a mathematician studying logic, your perspectives are generally limited to proof theory, recursion theory, model theory, symbolic logic, set theory, and category theory. Set theory is foundationally necessary for any math, but beyond that, I don’t find it as compelling as I had once hoped, when first reading about Godel’s theorems. Model Theory, also, I had hoped to find beautiful, but sadly it’s really an odd way to axiomatize a limited number of interacting structures. Because the axioms are themselves so limited, the theorems of model theory are also fairly limited, which is why it isn’t much of an active research area.

By contrast, I can’t say enough good stuff about Category Theory. Starting from the late 1800s, mathematicians began proving theorems that would eventually form the basis of Algebraic Topology, in my opinion perhaps the most elegant and difficult field of mathematics. The foundation of Algebraic Topology is a mathematical machine called “homology”. For years, mathematicians knew the value of “triangulating a surface”, where you cover a surface with triangles that overlap on edges. You can then, for example, count the number of triangles you’ve drawn, subtract the number of edges you see, and add the number of vertices. This gives you a number called the Euler number, or Euler characteristic. Interestingly, it turns out that the Euler number characterizes surfaces up to the “number of holes”. For example, the Euler number of a flat surface (say a sheet of paper) is 1. The Euler number of a sphere is 2. The Euler number of a torus (doughnut) is 0. It’s useful to be able to distinguish between surfaces, and the Euler number gives a way to do this. So mathematicians decided to take this concept to the next level, and try to distinguish between ANY space, regardless of dimension, in this fashion.

They built a complicated machine called homology. The idea is more or less as follows: take a triangle of any dimension. (For dimension 3, this is a tetrahedron.) Look at functions from this triangle into the space you’re considering. Mix these functions together, and “divide” out by the relationship obtained by the fact that triangles need to be glued to each other. This gives you a mathematical object called a group. Now if you do this for all dimensions at once, you get a whole lot of groups, one group for each dimension. These groups, it turns out, tell you a lot of information about your space, and they interact very nicely with each other, AND – mystery of mysteries – they’re easy to compute, much easier than you might imagine from this description.

This is where the magic comes in. Mathematicians had worked for years to develop the homology machine, and it turns out to have very simple properties. In fact, they were able to prove that even though they had to construct homology using functions defined on n-dimensional triangles, homology could be characterized by 5 very elegant axioms.

This was a stunning example of an emergent property in mathematics. The same way humans, and cognition, are properties that are fundamentally just due to the interactions of atoms, these properties really are best understood on a higher level. In some sense, this higher level of understanding “emerged” from complicated interactions on the lower level of physics. Well, math behaves the same way. Sometimes if you make the right sort of complicated construction in mathematics, properties emerge, and while those properties can be understood from a lower-level point of view, it’s much more beautiful, and elegant, and useful, to understand them on the same level.

Enter Category Theory. Mathematicians needed a general framework for understanding emergent properties, for understanding how some mathematical machines are just complicated instances of beautiful general behaviors. Category Theory is precisely how you describe this emergent behavior. It turns out that homology is nothing more than a “constructible functor” on the appropriate categories. The 5 axioms that define homology are defined in terms of category theory. The mess with functions and triangles is just like the atoms whirring around inside us – they make the theory in some sense, but they aren’t a useful perspective. They were just a way to generate the beautiful emergent properties that we cared about all along.

This relates to logic and your questions, Ed, because it turns out that one of the most useful and natural ways to understand logical systems is highly analogous to how one understands homology. If you think abstractly – much more abstractly than Model Theory would have you do – a logic could be considered as nothing more than

1. A collection of objects that you can consider “sentences”
2. A collection of implications between objects. If you have objects A, B, and C, you might have the implication A->B, but NOT the implication A->C. You can call these “morphisms” if you want to use the language of category theory.
3. A way of mixing sentences to generate new sentences. We can call this AND. A AND B gives you a new sentence.

Now you want to be able to assign these sentences truth values. Generally these values would be True and False – however, fuzzy logic and probability are fields that are very useful in risk assessment and computer science, so sometimes we want our truth values to be able to be weird. So abstractly we say that our truth values are “any category”, and that the assignment of these truth values must take values in this new category. We want the assignment to respect our implications, so if A is true, and A->B, we want B to be true. This simply requires that our map be a “functor” in the language of category theory. We also want our mapping to preserve the AND structure – this means our map is a “monoidal functor”.

Even if the terminology is confusing, the point is that logical systems are captured almost trivially by the elegant and descriptive language of Category Theory, in a much more broad and applicable sense than Model Theory ever used.

I can’t recommend Category Theory highly enough for those who are logically and technically minded. A good introductory resource (if you have some mathematical experience) is this free online book:

http://katmat.math.uni-bremen.de/acc/acc.pdf

Hope this ramble makes some sense!

the nature of mathematics is based on shape , form

and the form is based on the physical objects

Welcome back, twiffy.

Hi Twiffy,

When I first saw you post, I said to my wife, “I went fishing in the deep blue sea and caught a whale”.

As you probably know, I am very biased towards mathematicians, and your post was like great poetry to me.

I liked how you transitioned from homology to category theory which, unfortunately, I have not studied.

The reason that I mentioned Model Theory is primarily because I know a little about it, and I know that Wiki claims Alfred Tarski to be one of the four greatest logicians of all time.

I am excited to check out your references on category theory. Bearing in mind that I am a senile old man I may ask for help periodically. (Hope you don’t mind?)

Can you give any specific examples of how category theory is better than say bivalent classical Boolean logic as a stand alone subject matter?

Thanks Ed

I would like to follow up to my post on how the empirical world relates to mathematics, which was very conventional. This post on how mathematics relates to the empirical world is not conventional.

How does the mathematics relate to the empirical world?

Here I will build a specific relationship. It should be very intuitive.

We will let N be the cardinality of the collection of all apples.

Now we will define f such that f maps the Counting numbers from 1 to N to the empirical world as follows:

f(1) is defined to be 1’ apple *
f(2) is defined to be 1’ apple and 1’ more apple or 2’ apples
f(3) is defined to be 2’ apples and 1’ more apple or 3’ apples
.
.
.
f(N) is defined to be N’ less 1’ apple (defined by the preceding line) and one more apple.

This is a standard recursive definition where each line is defined by the line before it; and the first line is defined as f(1) is 1’ apple.

In previous posts I was using the terms one, two, three… to refer to the adjectives describing how we experience the empirical world. Now I have switched to the terms 1’, 2’, and 3’ to describe the same terms. This was simply because I did not have a good way to express the nth adjective of an empirical object.

1’, 2’, 3’ …N’ should in no way be considered numbers!

It should be clear that each number, 1, 2, …N is uniquely associated to 1’ apple, 2’ apples, … N’ apples.

We can define an association of objects in the empirical world, which we will symbolize as follows:

Without loss of generality we can assume that n < m. If not we just change the names. In the case that they are equal we know that n + n is less than N. But if n < m, then n + m < m + m < N.

We can go about showing that this association has all the properties of our standard addition.

This is the standard definition of addition in the Peano axioms.

Additionally, again assuming that m + m < N and n + n < N we have:

And assuming that n + m + l (the letter l not the number 1) < N we have:

I am pretty sure that the only people that would think that this is odd are philosophers!

The most striking way that I can put this is:

The rules of our mentality constructed abstracted mathematics (as long as the results of our operations are less than the cardinality of our empirical objects) [size=150]govern[/size] how this part of our constrained empirical world works!!!

For example, if we intellectually determine that 3 + 7 = 10, then three apples plus seven apples will be ten apples!!!

One of the straightforward questions is: are the abstracted Counting numbers finite? After all there are only a finite number of apples!

The answer, it seems to me, turns out to be cultural. (Oh yuck!!!)

Historically the empirical world was considered infinite, and if we were considering all empirical objects (the stars, the number of colors and others) the answer is clearly no - the Counting numbers are infinite.

On the other hand, our current models (after Quantum mechanics), at least in Western culture, of the empirical world indicate that it is finite. Perhaps the abstraction, if n is a number then, n + 1 is a number was premature?

Regardless of whether or not we can abstract the statement “if n is a number, then n + 1 is a number”, the rule has been codified.

A remaining question might be:

If the abstracted numbers can determine how the empirical world behaves, then can the axiomatic numbers do the same thing?

The answer is yes, in the same restricted manner as the abstracted numbers.

It turns out that the axiomatic numbers behave the same way as the abstracted numbers. Technically it requires that the axiomatic numbers to be a homomorphism** of the abstracted numbers. I sketched out this type of homomorphism in a play I wrote located at:

viewtopic.php?f=10&t=173698

The only difference is that I compared the Formal (set theoretic) axiomatic system to the Peano system. The reader can simply substitute A (representing abstracted mathematics for P to see how the Formal axiomatic system is homomorphic to the abstracted system.

• n’ apples represents all groups of apples whose cardinality is n. This is because we ultimately need to define an operation on apples that is good for any combination of apples with a given cardinality.

More rigorously n’ apples is defined as an equivalency class determined by the relationship R such that if a and b are subsets of the groups of apples then a is related to b by the relationship R if and only if the cardinality of a = cardinality of b.

To verify that R is in fact an equivalency relation we need to show:

aRa for all a. The cardinality of a subgroup of apples is uniquely defined
If aRb then bRa. If the cardinality of two subgroups of apples is equal then the order of the relationship R is not important. (This is called abelian or communitive in higher algebra).
If aRb and bRc, then aRc ‘ the relationships are transitive in nature.

Basically R is the same as equals except that the objects are being compared with regard to a specific property.

**Homomorphism Definition:

f(a* +F b*) = f(a*) +P f(b*) where f is a uniquely defined function on every element of its’ domain and range. Here the +F is addition on the formal numbers and +P is addition on the Peano numbers. It should be obvious that the domain and range can be completely different sets. Or in philosophic terms the domain of f can be ontologically different than its’ range.

Not in the slightest! Always happy to talk about fun math.

Well, you know, the study of classical boolean logic has been damn productive in computer science and other areas, and I wouldn’t want to claim that category theory is objectively better. However, aesthetically, ok fine, it’s better. In an abstract sense, category theory completely subsumes boolean logic. For example – and this might not mean anything to you right now, but bear with me – check this out.

C is the closed category whose objects are logical sentences, the morphisms are logical implication. T is the category whose objects are {False, True} and whose nontrivial morphisms is only False → True. A truth valuation is a closed covariant functor F: C → T.

This is a very simple categorical representation of all of boolean logic. When you study category theory in GENERAL you implicitly learn all of boolean logic, and so, so much more. Here’s a pretty accurate analogy: category theory is to boolean logic what the mathematical field of algebra is to high school algebra.

Hi Twiffy,

I could not bring up the following reference:

katmat.math.uni-bremen.de/acc/acc.pdf

And I would like to know more about category theory. Are there any other references? I do not mind spending the money on a good textbook.

I did however read the Wiki article and it seemed pretty straightforward. I particularly enjoyed the comments about analogies.

As far as I can tell, the axiomatic numbers from one to N, the abstracted numbers from one to N, and my constrained empirical objects, where N is their cardinality, form a category with my truncated definition of addition. (I am curious about your opinion on this matter).

My remaining comments will be on logic as a stand-alone subject. As usual I would appreciate any feedback.

Hi to All,

In Predicate Logic we can evaluate sentences using variables such as

1. There exists x such that x is a mammal.

2. There exists a number r such that 3 = 5r.

Or similarly:

1. There exists a number r such that r times r = -1.

The problem is that while we might be able to evaluate a variable in a manner that would make these statements true, we might feel increasingly uncomfortable, respectively, with those answers.

Horses, as well as many other examples, will evaluate statement 1 as true. But if we were working in an environment that requires whole numbers, like the Natural numbers, we might feel that statement 2 is not a true statement for us.

Additionally, most people, in my experience, can not actually use an imaginary number for real life experiences and would claim that statement 3 is false.

This means that our answers could be more useful if the variables and in fact the questions themselves were constrained to some context.

The Wiki article on logic talks about a Domain of Discourse, but I think that this should be made explicit. I do not know exactly how this is formally done (or at least how Tarski does it), but I think that the Domain of Discourse should mimic Model Theory.

We are clearly talking about sentences, but the sentences need to be formed from, and grammatically correct with regard to, an explicitly defined domain. Usually this domain can be specified as a set, but in any case the topic should also specify any valid functions and/or relationships operating on or in that topic.

For the mathematicians:

Model theory was an out growth of advanced algebra, which dealt with increasingly more complex operations on things that were similar to the Integers (though they could be finite sets as well), the Rationals, Algebraic numbers and the Reals. These were things like Groups, Ideals, Rings, and Fields. I don’t remember which mimicked which, but elementary advanced algebra was recognition of the fact that we can not always use all of the operations that are theoretically available to us. However, as is often the case with mathematics the concepts could be very generally applied.

When sentences that are structured with respect to a given set and operations or relations on that set, and are proven to be true, then they are said to provide a model of the specified set, operations, and relations.

In any case I think that it is clear that the Domain of Discourse should have such a structure or something similar to it. Otherwise logic will be dealing with true statements that will not relate to any given environment.

Personal note:

I feel that relationship between mathematics and logic could have been better presented, but I hope that the reader sees the need for more clarity with regard to the formation of logical assertions.

Thanks Ed

P.S. I think that Twiffy’s comments are very relevant to this topic.

Hi Ed,

http://katmat.math.uni-bremen.de/acc/acc.pdf