Mathematics

Mathematics is not easy to define. As with many other things, one can find a way out by abstracting out one level, to the process of trying to obtain a definition itself. One can attempt to describe his cognitive processes in searching for a definition. Although this doesn't actually answer the question, it does confine the uncertainty of the question into a structure, and one can somehow feel that he has a grip on the uncertainty in a way that he didn't before.

Many intellectual disciplines have names which refer to both the human practice of searching for knowledge and the actual knowledge or area of study itself. The suffix -ology has the meaning of the study of the subject, but by ‘biological material’ we do not mean biologists and biology textbooks. We should bear in mind this distinction when defining ‘mathematics.’ It refers to both what mathematicians do and also to the objective body of knowledge they are studying.

Mathematicians in the past and present may not be able to give you a definition of mathematics. They act, but do not know they act. Let us describe their actions, cleaving our knowledge of truth and reality between what can count as mathematics and cannot. This cleaving is an imposition of structure, and there may be hazy bits at the edges which were not sure about. As with all theory it should be viewed progressively (we are open to changing it) and reflexively (it may be worth applying it to its own derivation).

The experience

Let me attempt to describe mathematics as done by mathematicians. As I have said, it's not easy to define. Different people may have different ideas about it, and the definition may progress throughout time. For example, I would say that many people think of mathematics as the study of the natural numbers (1, 2, 3, etc.) and how to add, subtract and multiply them. They no more view the set of natural numbers, functions or proofs as mathematical objects than they view mathematicians or coffee cups as mathematical objects. If it is so obvious what a ‘mathematical object’ is, then there should be no debate about, for example, whether objects obtained via the axiom of choice ‘exist.’

If we divide reality into mental and physical, then mathematics is mental. If we divide the mental into external, sensations that seem to be a direct consequence of the physical, such as indigestion and hearing sound; and internal senses which don't, then it is internal. Again, we could divide the internal mental, and exclude headaches, fear and memories of past events. Nonetheless, we can reach a point where we feel we are getting to the bottom of the question of definition. Is ‘mental visualization' part of mathematics or not? I would say yes and no. At least we have now found somewhat the realm of experience in which mathematics is carried out.

There are further constraints that I can lay upon this realm of experience. Mathematical threads of thought are predictable. An individual could go through a thought, and have a feeling of recognition about the process. This leads me to a question that I haven't had before. Is humming a song to oneself a mathematical activity? When viewed as a source for musical enjoyment, it isn't. However, mathematical reasoning involves visualisation and the actual forms visualised may vary for the same underlying mathematical truth. If each note in the song is viewed as a member of a set, then thinking about the song is perceiving a mathematical object. Viewing a sequence of colours, letters or numbers would be equivalent.

Someone can do mathematics on their own, but it is still interesting to consider the communication of mathematical thoughts. One person could have mathematical thoughts, and explain them to an other. The other would have images in his mind which would progress according to his nature. Then they could disagree on some consequence of the initial idea. Perhaps he hadn't explained it well enough and they were thinking of different things. This misunderstanding could be resolved with more communication. But how do we really know their thoughts are aligned? What does it mean for them to be aligned? Their experiences of their own thoughts will be surely different. This leads us to an important point. All mathematics is up to isomorphism. Even if there are other elements, like colour, position and sound, or even emotion (such as nostalgia for the occasions in which one learned this area of mathematics) in one's mental experience, the mathematical nature of the object only extends to how it can be manipulated, so if the thought processes of two individuals progresses in an isomorphic manner, with the manipulations not involving the extraneous sensations, then when they communicate they will agree on the result.

The meaning

We would like to think that if we could strip away all of the superfluous thoughts we have when we think about mathematics, then there is a deep mathematical truth below it.

Biology

People imagine that they can do mathematics as if their mind was a collection of Platonically smooth, perfectly interlocking crystal gears. That may be the ideal, but we should consider whether that is actually what is going on. Consider the process of counting, as carried out by a brain. It is a lot of atoms acting according to physics. We can't understand all of those, so instead we make a theory to jam the atoms inside. We have constructs like neurons, axons and dendrites. With a higher-level theory we might have thoughts, feelings and memories. At this point in the description I start to freak out. I thought I was just imposing a structure which didn't actually reflect physical reality, which was actually based in the atoms. However, the brain has a consciousness, and the nature of that consciousness is similar to one's high level theory! Counting up from 1 to 100, the atoms in the mind might randomly move out of the pattern you had put them into, and then the consciousness would have to disappear too, or change in nature. For example, if your structure would invariably make no mistakes in counting, then the actual consciousness might skip a number on the way. In fact, people do make mistakes when adding numbers in their head.

Nevertheless, people do have souls, and they are only so complicated, supposing physics has a finite complexity, and basing brains in physics. If a soul is a finitely describable object, then it might be possible for a mind to sharply and directly comprehend a mathematical truth, or to manipulate forms with no chance of error.

Even the sharpest image could be sharper. I listen to a lot of low-quality music on YouTube, but now and again I hear a high-quality version, and I'm astounded that I would be capable of experiencing such sharp experience. I imagine that there could be experiences where the world is seen and heard much more sharper. Entering such a world we would be ecstatic. Maybe the ecstasy would fade a bit after a while, but the sharpness would still be there to be enjoyed, and we wouldn't claim that the enjoyment of the sharpness was merely relative to the previous lower level of sharpness.

Specifics

We described the mental experience of mathematics as deterministic progressions of thoughts, which have meanings. These meanings may be applied to analyse the universe, to describe the potential progress in time of space and matter and consciousness.

I have not so far actually given a description of what mathematical objects and process that one may think about. Not doing this allows us to contemplate the possibility of mathematical truths that meet our qualitative definition (meanings of classes of deterministic thoughts that are without regard to physical sensation or emotion) but that we are completely ignorant of.

As far as we are concerned, we can restrict our attention to sets, as all objects of interest can be built out of sets, and operations of interest out of set operations, I think.

Sets

I still don't know what a set is. Modern mathematics seems to say that a set is something in the cumulative hierarchy of the Von Neumann universe. What's this hierarchy? A collection of sets of sets indexed by the ordinals. What are the ordinals? Equivalence classes of well-ordered sets. Oh wait, what's a set again? We're going round in circles here.

Either that, or an element in a model of ZFC. Oh wait, a model is itself a set, but the collection of all sets can't be a set. What's going on?

I am yet to find a satisfactory explanation of the question of what a set is. Using the cumulative hierarchy as our definition, the question is the same as asking what an ordinal is. I thought I had an way to do it, but it turned out that I was wrong, if I remember correctly. I think we'll have to use an intuitive idea of a deterministic mental process, but I haven't looked into this much.

So, the story goes that after mathematicians discovered all these paradoxes of ‘naive set theory,’ they developed ZFC and put mathematics on a firm footing. It tells us what we can and can't do, so no silly objects like ‘the set of all sets’. This is clearly false. If ZFC is consistent (which I would say it was), it is not complete, so doesn't have a single model, but multiple models. It thus fails in giving a definition of ‘set,’ because there's a different definition for each model.

Models of ZF are confusing too, being sets but not really, because they're ostensibly the ‘set’ of all sets. Let's forget about models, and just say a set is ‘something that you can prove to uniquely exist using the mechanics of ZFC,’ and I think we get a bit closer to the truth. This tells us that we can do power sets, unions, separations, and so forth, and restricts us from attempting to talk about objects that don't exist.

That's not the whole story. It only gives countably many objects, but as we know, R is uncountable. Even the objects accessible to human reason are uncountable. The diagonal process is constructive, so it is no good saying that we could find some way of listing all accessible real numbers, because the new number would be accessible too. (Here human reason is taken in a broad sense. The human species may not exist forever and if it does, might only do a countable number of things. I am talking about ‘potential thoughts of human-like reason.’) I think that most mathematicians would have no problem talking about ‘the set of sets proven to exist by the axioms of ZFC,’ even though, clearly, that set isn't proven to exist by ZFC.

If we could define a set (or an ordinal) we would be almost there. We could just define ‘mathematics’ as the study, by mathematicians, of these abstract, intuitively defined objects. Talking about objects which aren't sets would be a trivial modification of the intuitive definition. I feel that this is a genuine question! It must have an answer. Honestly, though. What is a set? It shouldn't be hard to answer. We'd like to define a set, then, using some kind of intuitive principle. Let's have a go:

  • The empty set is a set
  • Some other stuff, maybe power sets or unions
  • Given any mental process that churns out sets in a deterministic way, there is a set whose members are the churned-out sets.
  • Nothing else is a set.

I don't know how to define ‘mental process’ and ‘deterministic,’ though.

The foregoing fails to give us any uncountable sets. It could be worth defining ω1 to be the collection of countable ordinals defined in a similar way, though.

The bewildering and ungraspable potential for repeated generalization can be found in many different circumstances - in trying to express the biggest number one can think of, or in writing down ordinals, or large classes of sets. It can be applied to any body of knowledge. First we have knowledge, then we have knowledge of the process of our knowing (‘meta-knowledge’ if you will), then knowledge of this derivative knowledge, and so on. Then we have knowledge of this collection of forms of knowledge of size ω, and so on up into the ordinals. You might call them ‘knowledge,’ ‘meta-knowledge,’ ‘meta2-knowledge,’ ‘metaω-knowledge,’ ‘metaε0-knowledge,’ and so on.

We can talk about layers of knowledge indexed by the ordinals, but until we actually populate them with knowledge in which we are interested (which takes time) these layers are empty. The idea of these is only useful insofar as they might make us think of researching more generally, but we can only really do that once we've built up a good amount of understanding and interest on one level. Just because we can write down bigger and bigger ordinals without end, it doesn't mean that we have understood in the fullest sense the manifold levels of knowing of any body of knowledge.

I feel that I am able to describe (to some level of satisfaction) what the ordinals are that human beings are capable of understanding. To do this, we view the human mind as something that is capable of spotting patterns. In the phrase ‘spotting patterns,' there is much meaning. I think I hit on a good thing before with a qualitative definition - methods by which situations may develop. (I may have to learn about mathematical algorithms, and what precisely the relationship is with the ordinals is. Ordinals may not strictly be methods of development, but they do feature in temporal mathematical thoughts, so I don't think I've gone too far off track.) Depending on your philosophical bent, you might think of them as the Platonic ideas. I can't give a more precise definition. As I say, all I can do is restrict myself to what human beings can do, as anything else can't be understood, as it is impossible for a being to completely understand itself. Patterns are spotted by the thought, not just in the outside world or in perception, but also in thought itself, and this is how we come to think generally. It is impossible for a human being to completely explain what patterns a human being may find, as then he would have substantially understood himself.

Russell's paradox - not a problem at all

What is a set? A naive answer is ‘a collection of mathematical objects.’ We think we know what this means, but unfortunately there is an assumption in that answer that is easy to miss. Likely you didn't think of the set being a member of itself. When do things, like bags or boats, contain themselves in the real world? To be more precise, we would have to exclude self-membership when we say what we think of by a set. There are other questions to consider too, like whether the set can be a member of a member of itself. I'm not sure but I would guess not. If you allow such things (set theory without the axiom of foundation is a genuine field of study, I hear) you would have to be careful to exactly specify what it was you were thinking about.

One of these paradoxes is Russell's paradox. This is as follows: Consider the set of all sets that are not members of themselves. Then is it a member of itself? If it is, then it isn't. If it isn't, then it is. This a contradiction. This paradox shows that there are problems with the naive definition above.

There are two explanations of this paradox and why it shouldn't trouble us. Firstly, it is an equivocation fallacy, and secondly a problem with natural language.

First, the equivocation fallacy. We are using two different intuitive ideas of set:

  1. A collection of mathematical objects
  2. A function from the class of mathematical objects to the set {out, in}

The set of all sets surely exists, we think. Just take the function to be constant with a value of ‘in’. If we had mathematical objects which weren't sets, we can fix it in the following way.

Given a collection of objects, we could imagine going through it, selecting which objects we want to keep, according to whether the object possesses some property, and thereby obtaining a new collection. Moreover, even if the collection is infinite, we could produce an object that behaves as a collection, that is to say, we can definitely say whether a given object is a member of it or not, in the following way. Given an object, we first look to see whether it is a member of the original set. Then we see whether it possess the given property. (This process is called the ‘axiom of separation.’)

Given the set of everything, perform the foregoing with the question ‘Is this object a set?,’ giving the set of all sets. Do the same with the question ‘Is this set a member of itself?,' giving the paradoxical object which is the object of this paradox.

The problem is that we've applied both 1 and 2 for our idea of a set. If we only use idea 1, we abandon the idea of a set being a member of itself. In this case we know that there is no set of all sets because such a set would contain itself. If we only use idea 2, there is a set of all sets, but the axiom of separation doesn't work. We know it doesn't work because of Russell's paradox. We can also consider y = {sets x | x in x}. Is y in y? There's no way to answer this because the definition of y ends up referring back to y.

The foregoing shows us that if we want to define a mathematical object in terms of other mathematical objects, those mathematical objects cannot be defined in terms of the mathematical object we are trying to define (not even indirectly). There is a temporal aspect to this way of thinking about mathematical objects. First we have some mathematical objects, and afterwards we create or define some new ones in terms of the old ones.

English grammar allows noun phrases that are meaningless. There is no ‘current king of France.’ There is no single object referred to by the phrase ‘the integer that squares to 4,' as both +2 and -2 square to 4 and are integers. The grammatical form ‘the' + noun phrase without 'the' in front of it is capable of misleading us. The situation is similar with the phrase ‘the integer which squares to 5.’ We should also accept that the form ‘the Object class that Proposition' is sometimes meaningless in the same way.

Complicated and class propositions

The conscious human mind is only so large. We can only visualize so many objects, mathematical formulae of a certain length. We rely on our memories and paper to store that which we can't hold. The totality of the proof of most mathematical propositions cannot be comprehended all at once. The human brain is capable of comprehending some mathematical ideas, like the pigeonhole principle, and visualizing some mathematical objects (like some simple graphs) and visualizing the elements of an algorithm that would yield another (like the expression ‘999’, which represents a number that can't be imagined, but we know we could count to 999, or count out 999 objects or do something 999 times, by using the counting down algorithm we believe to exist in our memories). The ideas we can totally comprehend I accept as totally true. These are visions of the mind, which I believe is capable of perceiving truth.

However belief in statements about more complicated entities rely on our belief in the reliability of our memories or the constancy of the paper we've written on, or are reading an argument from, not to change from one thing to another when we refer to it; and are therefore less certain. In this statement we have imagined an analytical process which is more complicated than we can imagine, but nonetheless believe that this reasoning process exists, and furthermore the events of time continue beyond what we can take in in one thought. Therefore I am not adopting a position which says that some small integer (4,5,6,7 perhaps) is the biggest. Statements about complicated entities, let's call them complicated propositions, are not therefore meaningless. Even if there are conscious trains of thought which experience false train-tracks of reasoning, we can in one thought imagine that there is such a thing as a correct train-track, that it can be followed, and that it would reach some definite result.

We move onto the next stage. We had postulated some small objects, like 0, 1, 2, 3, viewed as von Neumann ordinals being representatives of sets with that many objects within. Now we can work with arbitrarily large objects, as long as they are the end result of some mental process. An example of this is that we can talk about large integers. These are ‘the things you can get to by repeatedly taking a successor to zero (with no loops)". We are now happy to make statements like ‘999 times 998 = 997'002’, which can be interpreted in two ways: manipulating the symbols ‘999’ and ‘998’ in a certain way will give you the symbols ‘997'002' (which even if you can do in your head, you may not necessarily be able to perceive the whole process of the multiplication algorithm in your head all at once); or even better for demonstrating this stage, starting at zero, and performing the process of adding 1 999 times 998 times yields the same number as starting at zero and adding 1 997'002 times, which is a totally incomprehensible proposition. It is inconceivable that anyone could visualize 997'002 objects in their head all at once.

We move onto the next stage. We have an idea of what it means for a mathematical object to be an integer. Now we can talk of the class of integers (we shan't use the term set just yet). Assuming rationality, assuming that the trains of thought of mathematical reasoning will be correct, we can ponder meaningful statements true of the class of integers. Here we get our first glimpse of infinity. We can make meaningful statements about infinity, like ‘no two odd integers can be added to make an odd integer.’

What we have just done is take an idea of a certain type of object, and then considering all objects of that type, calling it a ‘class.’ With some finite set (I haven't defined what a set is yet, but a finite set (the general definition of which can easily be thought of in a single thought) is included in the objects we allowed two paragraphs ago) we can do this with the idea of a subset, thus obtaining the power set. As long as the idea is clear, a class can be spoken of. For a while I believed that the power set of the integers was not as real as the integers, because we'd included subsets like ‘for each integer, toss a coin to see whether it's in our subset’ which were impossible to mathematically define. However, an arbitrary subset is about as real as an arbitrary number. Statements about an arbitrary member of a class only say that should we think of an actual member of that class, the statement would be true of it. Therefore power sets are worthy of our consideration and statements about them are meaningful. (I also question what it means to ‘toss a coin.’ Such an object would be reliant on someone in the real world tossing a coin. The results might be definable by some complex mathematical formula if the universe is predictable, but the person would get bored eventually. The only point in favour of the integers being more real than its power set is that if the universe goes on forever in time, there will a moments of time corresponding to every integer, and therefore the integers have a physical manifestation; but it's hard to think of a corresponding manifestation of the power set.)

Part of the reason why I have gone through much of the reasoning above is that I want to get some grounding for trying to understand some paradoxes of set theory, like Russell's paradox, Richard's paradox, Cantor's paradox, and the Burali-Forti paradox.

Richard's paradox

This occurs when applying a diagonal method to verbal descriptions of numbers.

I can't be bothered to go into much detail at this point in time. However, I will make a few remarks.

First of all, I believe it is easier to think of somebody else contemplating descriptions of numbers, rather than considering ourselves doing it. I concede that this may not be necessary; nonetheless, it stops ourselves tying ourselves in knots.

What is denotation? A concept of a class of physical objects is a mental structure that operates by being metaphorically lain aside phenomena; and, by the level of similarity between them, the decision is made whether the phenomena belong to the class. Concepts of mathematical objects are not quite the same. Let's be conservative and only define a concept of a real number (to allow the paradox to work). We take the concepts of the decimal representations of real numbers for granted. (It is certainly interesting to go into more detail what one of these is like, but we can leave that aside for now. In short, these are ways of giving any digit we require of the decimal representation. I use the word "representation" here, but that is not to say that these concepts cannot be taken as primary, as if they are representing something else.) A concept is said to denote a particular real number if it can be "lain aside" one of these concepts and it can be seen to give the same decimal representation. (Are these not one and the same concept, and not two separate concepts, one may ask? No, because these are two separate pieces of "mental machinery," which happen to give the same result.)

These definitions posited, the paradox can be easily resolved, thus: the diagonally generated concept fails to denote any real number.

However, it may indeed denote a number to a higher being observing the agent's thought processes. We think a bit further, and come up with the following putative description of a paradoxical concept: "A number defined by a diagonal process which considers all higher observers." I haven't followed this line of thought all the way through yet, but I don't think it will be hard.

Miscellany

Continuum hypothesis

I believe the continuum hypothesis has a truth-value. It states that the cardinality of the collection of all countable ordinals is the same as the cardinality of the power set of the natural numbers; or in other words there exists a bijection between the two sets.

It's been proven that this cannot be proven in first-order logic using the axioms of ZFC. Does this mean it is false? We might argue that because we can't prove the existence of a bijection, we cannot exhibit a bijection, and so no bijection exists. However, the same would be true of a set of intermediate cardinality, which it seems that we can't exhibit either, and the non-existence of such a set has the opposite implication as to the truth value of the hypothesis.

But surely: either such a set exists, or it doesn't! Either such a bijection exists, or it doesn't!

Of course, we'd have to know what a set is, what a mathematical object is, and what 'exists' means, which I don't.

Axiom of choice

I think that it's true but not obvious, but I'm open to it being false. I view a mathematical object as something which can be comprehended. In any case we come across, it seems that we could indeed find a choice function. I need to see why this is.

Well, obviously it doesn't exist for P(P(N)). But in more limited cases, say a definable countable subset of P(P(N)), maybe it does exist. I don't believe that it matters that much, or that you could prove something concrete about the naturals that was actually false using the axiom.

Graphs

Graphs as presented in a course on graph theory are not mathematical objects. A graph is defined as a set V (the vertices) along with a subset E of V(2) (the edges). But all the graphs considered, like K4, aren't fully defined. V could be any set with four elements. Instead, K4 should be viewed as a class of graphs.

Couldn't you make exactly the same argument with, say, the whole numbers? Why am I even bothering to make this point? The axiom of choice is illustrated with infinitely many pairs of shoes and infinitely many pairs of socks. It's alleged that there's no canonical way of choosing a sock from a pair. However, when do objects like pairs of socks ever appear? A pair of socks is like E2, an empty graph on two vertices. As I said before, that's not a real object, so we don't need to worry about it.

Rewriting rules and deduction systems

I believe that manipulations of strings of symbols from a finite alphabet with a finite number of simple pattern matching rules are hugely important for modelling not just mathematical reasoning, but human reasoning. (For example in variants and transformations of grammatical constructions.) (And similar stuff like symbol trees, with the possiblity of subtrees being mapped to each other using more specific rules. E.g. subject-object inversion such as "Der Mann hat den Hund getreten" to "Der Hund hat den Mann getreten". Note these examples could be completely wrong.) One is contained within the other.

N is the typical example of a countable set. Consider P(N) - can we define a member which is unattainable to human intelligence?

I have a strong hunch that any line of human reasoning whatsoever can be expressed using these finite deduction systems as above. We want to consider the subsets of N, but we hit a big problem. How do we define "subset"?

Gödel's incompleteness theorem shows that this is a problem. It shows that we cannot define a set (nor, consequently, other classes of mathematical objects) using merely a list of first-order logic formulae.

Suppose we define a subset as a method that a human being can use to say whether a number is in the subset or not. According to my hunch above, any such method could be expressed as a series of modification rules, starting with the number represented in some format, with exactly one of the symbols "<in>" or "<out>" being among the consequences for all numbers.

We can easily apply a diagonal method to the set of such rule sets to define a new subset which is not defined. We would list all rulesets and restrict ourselves to the ones which meet the criteria above, i.e. which define subsets. However, to actually use this to work out whether a number is in it or not would probably require violating the halting theorem, and there is no method we can use to see which rulesets define subsets. In that sense, it is not actually subset in the sense of "a method that someone can actually use".

It would violate the halting problem because we can map the aetiology of an algorithm to a rooted tree where each node has at most one child (representing the next time-step), and therefore the execution of algorithms is a sub-category of these general reasoning processes. We can represent both deduction systems like first-order logic and algorithms like matrix inversion.

  • This raises the question of whether there are actually algorithms for which humans cannot know whether they finish.

However, in so far as human reasoning is captured by these kind of trees, the consequences thereof should be able to be generated by computers.

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