I've wanted to write a series of posts here on logic and the foundations of mathematics for a while now. There's been some recent discussion about the ontology of numbers and the existence of mathematical entities, so this seems as good a time as any to start.
Many of the discussed philosophical problems, as far as I can tell, stem from the assumption of formalism. That is, many people seem to think that mathematics is, at some level, a formal logic, or at least that the activity of mathematics has to be founded on some formal logic, especially a classical one. Beyond that being an untenable position since Gödel's Incompleteness Theorems, it also doesn't make a whole lot of intuitive sense since mathematics was clearly done before the invention of formal logic. By abandoning this assumption, and taking a more constructivist approach, we get a much clearer view of mathematics and logic as a whole.
This first post is mostly informal philosophizing, attempting to describe exactly what logic and mathematics is about. My second post will be a more technical discussion accounting for the basic notions of logic.
Intuitions and Sensations
To begin, I'd like to point out a fact which most would find obvious but has, in the past, lead to difficult philosophical problems. It is clear that we don't have direct access to the real world. Instead, we have senses which feed information, even if dishonestly, to our mind. These senses may be predictable, may be potentially modeled by a pattern which mimics our stream of senses. At some level, we have direct access to a sensory signal. This signal is not a pure, unfiltered lens on the world, but it is a signal, independent of, but directly accessible by, us.
We also have access to our intuitions, the part of our thoughts which we may label "ideas". We may not have total access to all our faculties. Much of our mental processing is done outside the view of our awareness. If I asked you to name a random city, eventually you'd come up with one. You'd, however, be hard-pressed to produce a trace of how that city's name came to your awareness. Perhaps you could offer background information, explaining why you'd name that city, among all the possibilities. Regardless of such accounts, we'd still lack a trace of the signal sent by your consciousness ("I want the name of a city, any city") reaching into the part of your mind capable of fulfilling such a request, and the subsequent reception of the name into your awareness. We don't have such detailed access to the inner-workings of our mind.
It seems that those things which we have direct access to are, in fact, part of us. Those intuitions within our awareness, those filtered signals which we directly experience, make up our qualia, are instantiated in the substrate of our consciousness. They may be thought of as part of ourselves, and to say we have access to them is to say that we have direct access to those parts of ourselves of which we are aware. This, I think, is trivially true. Though that isn't essential for the rest of this piece.
We may distinguish normal intuitions from senses by the degree we can control them. Intuitions are controllable and manipulable by ourselves, while senses are not. This isn't perfectly clean. One may, for example through small DMT doses, cause one to experience controllable hallucinations which are a manifestation of direct (though not complete) control of the senses. Also, there are plenty of examples of intuitions which we find difficult to control, such as ear-worms. For the sake of this work, I will ignore such cases. What I want to focus on are sensory sensations fed to our awareness passively and those intuitions which we have complete (or complete for practical purposes) control. These are the sorts of things needed for logic, mathematics, and science, which will be the primary focuses of this series. For the remainder, by "sense" and "sensory data" I am referring to those qualia which are experienced passively, without deliberate control; by "intuition" I am referring to those intuitions which are under our direct and (at least apparent) total control.
At this stage, it's useful to make a remark about language and grounding. Consider what I might be saying if I describe something as an elephant. Within my mind is an intuition which I'm assigning the word "elephant", and in calling something presumed external to me an elephant, I am asserting that my intuition is an approximate model for the thing I'm naming. The difference between the intuition and the real thing is important. It is practically impossible to have a perfect understanding of real-world entities. My intuition tied to "elephant" does not contain all that I might consider knowable about elephants, but only those things which I do know. A veterinarian specializing in elephants would certainly have a more accurate, more elaborate intuition assigned to "elephant" than a non-specialist, and this wouldn't be the full extent to which elephants could be modeled. In essence, I'm using this modeling intuition as a metaphor for an elephant whenever I use that word.
Based on this, we can account for learning and disagreement. Learning can be characterized as the process of refining an approximately correct intuition modeling something external. A disagreement stems from two main places. Firstly, two people with similar sensations may be using differing models. From these differences, two people may describe identical sensations differently, as their models might disagree. Secondly, two people may think they're getting similar sensations when they are not, and so disagree because they are unable to correctly compare models, to begin with. This is the "Blind men and an elephant" scenario.
This account also cleanly explains why we can still meaningfully talk about elephants when none are present. In that case, we are speaking of the intuition assigned to "elephant". Additionally, we can talk about non-existent entities like unicorns unproblematically, as such things would still have realities as intuitions. An assertion of existence or nonexistence is really about an intuition, a model of something. The property of existence corresponds to a prediction of presence in the real world by our model, non-existence to our model predicting absence. The correctness of these properties is precisely the degree to which they accurately predict sensory data.
Intuitions need not be designed to model something in order for it to be used to model something else. If I try to describe an animal which I'm only the first time encountering, then I may construct a new model of it by piecing together parts of older models. I may even call it an "elephant-like-thing" if I feel it has some, if limited, predictive power. In this way, I'm constructing a new model by characterizing the degree to which other models predict properties of the new animal I'm seeing. Eventually, I may assign this new model a word, or borrow a word from someone else.
One can also create intuitions without attempting to model something external. If you were a mind in a void, without any sensory information, you should still be able to think of basic mathematical and logical concepts, such as numbers. You might not be motivated to do so, but the ability to do so is what's relevant here. These concepts can be understood in totality as intuitions, completely definable without external referents. Later, this will be elaborated on at length, but take this paragraph as-is for the moment.
Even if an intuition was created without intent to model, it still can be used as such. For example, one can think of "2" without using it to model anything. One can still say that a herd of elephants has 2 members, using the intuition of 2 as a metaphor for some aspect of the herd.
Some notion I've heard before is that it seems like a herd with 2 members would have 2 members even if there was no one around to think so, and so 2 has to exist independently of a mind. Under my account, this statement fails to understand perspective. It is certainly the case that one could model a herd of 2 using 2, regardless of if anyone else was thinking of the herd. However, even asking about the herd presupposes that at least the asker is thinking about the herd, disproving the premise that the herd isn't being thought about. If it were truly the case that no one was thinking of it at all, then there's nothing to talk about. The question would not have been asked in the first place, and the apparent problem then vanishes. It is clear at this point that stating "a herd has 2 members" does not make 2 part of our model of the world.
At this point, I will introduce terminology which distinguishes between the two kinds of intuitions discussed. Intuitions which are potentially incomplete, designed to model external entities will be called grounded intuitions. Those intuitions which may be complete and may exist without modeling properties will simply be ungrounded intuitions.
One common description of reality stemming from Platonism is that of an imperfect shadow or reflection of the transcendental world of ideals. After all, circles are perfect, but nothing in the world described as a circle is a truly perfect circle. By my account, perfection doesn't come into the picture. A circle is an ungrounded intuition. An external entity is only accurately called a circle in so far as the intuition of a circle accurately models the entity's physical form. The entity isn't imperfect, in some objective sense. Rather, the grounded intuition of that entity is simply more complex than the ungrounded intuition of the circle. The apparent imperfection of the world is only a manifestation of its complexity. Grounded intuitions tend to be more complicated than the ungrounded intuitions which we used to approximate the real world. This is, at once, not surprising, but significant. If we lived in an extremely simple world (or one which was simple relative to our minds) then we might create ungrounded intuitions which were simpler than the average ungrounded one. We may then have trouble distinguishing between sensory data and intuition, as all facts about the real world would be completely obvious and intuitively predictable.
Ontological Commitments of Ungrounded Entities
I think it's worth taking the time to discuss some content related to ontological commitments and conventions. Ontological commitments were introduced by Quine, but I won't hold true to the notion as he originally described it. Instead, by an ontological commitment, I am referring to an assertion of the objective existence of an entity which is independent of the subjective experience of the person making the assertion.
Let's take a scenario where two people are arguing over what color the blood of a unicorn is. One says silver, the other red. Our goal is to make sense of this argument. Assuming neither people believe unicorns exist, what content does this argument actually have?
First, it behooves us to make sense of what a unicorn is, and what commitments we make in talking about them. For the moment, I'll stick to a conventional distributive-semantical characterization of meaning (I plan on making a post about this quite some time from now). Through our experience, we eventually associate words like "blood", "horse", and "horn" with vectors inside of some semantic space. We can then combine them in a sensical way to produce the idea of a horse with a horn, a new vector for a new idea, a unicorn. When talking about commitments, we need to make a distinction between two things; commitments to expectations, and commitments to ideas. When we define unicorns in this manner, we are committing ourselves to the idea of unicorns as something that's coherent and legible. We are not making a commitment to unicorns existing for real, that is we do not suddenly expect to see a unicorn in real life. This may be considered an ontological commitment of a sort. We certainly ascribe existence to the idea of a unicorn, at least within our own mind. We don't, however, ontologically commit ourselves to what the idea of unicorns might theoretically model. Since all sentences cannot help but refer to ideas rather than actual entities, regardless of our expectations, the assertion that unicorn blood is silver pertains to this idea of unicorns, nothing that exists outside of our mind.
I'd like to digress momentarily to talk about this standard conundrum:
If a tree falls in a forest and no one is around to hear it, does it make a sound?
This question has a standard solution that I'd consider universally satisfactory. Ultimately, the question isn't about reality, it's about the definition of the word "sound". If by "sound" the asker is speaking of a sensation in the ear, then the answer is "no". If they mean vibrations in the air, then the answer is "yes". Under the distributional semantics of the word "sound", we can talk about this word having values in various directions. For some people, "sound" is assigned the region defined by a positive value in the direction corresponding to sensations in the ear. For others, "sound" is assigned to the region with positive value in the direction corresponding to vibrations in the air. These two regions have heavy overlap in practice. When we experience a sensation, it's rare for it to have a positive value in one of these, but not the other. And so, we assign one of these regions the word "sound", most of the time having no problem with others who make a different choice but arriving at disagreements over questions like the above.
But which is it? What does "sound" actually mean? Well, that's a choice. Consider the situation in detail. Is there anything that needs to be clarified? Are there vibrations in the air? Yes. Are there any sensations in an ear caused by these vibrations? No. So there's nothing left to learn. All that's left is to decide how to describe reality. It may even be useful to split the term, to talk about "type-1 sound" and "type-2 sound", which usually coincide, but don't on rare occasions. Regardless, it's a matter of convention, not a matter of fact, whether the word "sound" should apply.
And so, we're in sight of the resolution to the unicorn blood argument. One person has a region in their semantic space corresponding to one-horned horses with silver blood, and want's to assign that region the word "unicorn". The other person has identified a close-by semantic region, but there the blood is red, and they want that to have the word "unicorn". Note that neither would think that the others claim is nonsense. The argument is not predicated on, for example, one person thinking the idea of a unicorn with red blood is incoherent. Both parties agree that each other have identified meaningful regions of semantic space. They are making identical ontological commitments. What they are disagreeing on is a naming convention.
Throughout this series, I will often discuss mathematics and logic as fundamentally subjective activities, but this does not mean I reject mathematical objectivism as such. Rather, the objective character of mathematics moves from being an aspect of mathematics itself to being an aspect of how it's practiced. Mathematics is done as a social activity carried by a convention which is itself objective: or at least (ideally) as objective as a ruler. Showing that someone is mathematically wrong largely boils down to showing which convention a person is breaking in making an incorrect judgment.
Brouwer, who was the first to really push mathematical intuitionism, described mathematics as a social activity at its core. As a consequence, he argued against the idea of a formal logical foundation before Gödel's incompleteness theorems were even discovered.
The basic idea of constructivism is to limit our ontological commitments as much as possible. Consider the well known "I think, therefore I am". It highlights the fact that the act of thinking and introspection itself implies an ontological commitment to the self. Since we are already doing those things, it's really not much of a commitment at all. Similarly, the fact that I am writing in a language commits me ontologically to the existence of the language I'm writing in. As I'm doing this anyway, it's not much of a commitment. For this, I call these sorts of commitments "cheap commitments".
Mathematical and logical entities are ideas. By discussing them, we are committing ourselves to the existence of these entities at least as ideas. For example, if I say "there exists an even natural number", I am committing myself to the ideas of natural numbers and evenness. I'm also committing myself to the coherence or soundness of these ideas, that the statement in question is meaningful modulo the semantics of the ideas used.
I can easily make grammatical-looking sentences that seem to make some sort of expensive commitment. For example, I could say that g'glemors exist and that a h'plop is an example of a g'glemor on account of hipl'xtheth. If I said those things with any sort of seriousness I'd be committing myself to the existence of those mentioned things at least as ideas, as well as the soundness of those ideas. Being nonsense words not representing anything at all, I'd obviously be misguided in making such commitments, they certainly aren't cheap.
The point of a constructivist account is to describe mathematical and logical ideas in such a way that one is committed to their soundness in a cheap way. And here we can start to see the significance of characterizing mathematics and logic as being about ungrounded entities. In order for my commitments to those ideas to be cheap, they must be totally characterized by something that comes from within me, by something that I'm doing anyway when discussing those ideas.
Precommitments and Judgments
We say that an idea is a cheap commitment if, in defining the notion, we summon the entity being defined, or perform the activity which we are judging to be the case. In order to do this, we need to pay attention to precommitments.
A precommitment is a prescription we make of our own behavior. It's an activity which is being done so long as those prescriptions are being followed. Precommitments are the core of structured thinking. Whenever we impose any pattern or consistency to our thinking, we are making a precommitment. By analyzing our precommitments closely, we can construct, explicitly, ideas which are cheap ontological commitments. If we are actively doing a precommitment, then we can cheaply acknowledge the existence of the idea conjured by this precommitment.
Many ungrounded intuitions arise as a form of meaning-as-usage. Some words don't have meaning beyond the precise way they are used. If you take a word like "elephant", it's meaning is contingent on external information which may change over time. A word like "and", however, isn't. As a result, we'd say "and"'s meaning fundamentally boils down to how it's used, and nothing more. Going beyond that, if we are to focus on ungrounded intuitions which are complete and comprehensible, then we are focusing precisely on those ungrounded intuitions who's definition is precisely a specification of usage, and nothing more. That specification of usage is our precommitment. Of course, usage happens outside the mind, but the rules dictating that usage aren't, and its those canonical rules of usage which I mean by "definition".
The basic elements of definitions are judgments. Judgments include things like judging that something is a proposition, or is a program, or is some other syntactic construction. Judgments also include assertions of truth, falsehood, possibility, validity, etc of some data. However, be aware that a judgment simply consists of a pattern of mental tokens which we may declare. Regardless of what preconceptions about possibility, truth, etc. one has, these should be overwritten by the completed meaning explanation in order to be understood as a purely ungrounded intuition and a cheap commitment.
When we make a judgment, we are merely asserting that we may use that pattern in our reasoning. Precommitments, as we will make use of them here, are a collection of judgments. As a consequence, what we are precommitting ourselves to is an allowance of usage for certain patterns of mental tokens when reasoning about a concept. The full precommitment summoning some concept will be called the meaning explanation for that concept.
Ultimately, it is either the case that we make a particular judgment or we don't. That, however, is a fact about our own behavior, not about the nature of reality in total, in essence. Furthermore, someone not making a particular judgment is not automatically making the opposite, or negated, judgment. In fact, such a thing doesn't even make sense in general. As a result, we don't reproduce classical logic. Though, as we'll eventually see, there are constructive logics which are classical. However, it's worth dispelling the idea that there's "one true logic". Questions about which kind of logical symbols, classical, intuitionistic, linear, etc. is the "true" one are nonsense. One is only correct relative to some problem which has an element which is to be modeled by one of these. Whichever is the more accurate model is the correct one, there is no "one true logic", and it's certainly not the case that the intuitions which make up mathematics are governed by a classical logic. For example, the existence of theoretically unsolvable problems (e.g. the halting problem) illustrates that our capacity for judging truth is fundamentally constrained, not by some objective transcendental standard for truth, but rather by our ability to make proofs.
To summarize, to define a concept we give a list of judgments, rules dictating which patterns of tokens we can use when considering the concept. So long as these rules are being followed, the concept exists as a coherent idea. If the precommitment is violated, for example by making a judgment about the concept which is not prescribed by the rules, then the concept, as defined by the original precommitment, no longer exists. There may be a new precommitment that defines a different concept using the same tokens which is not violated, but that, being a different precommitment, constitutes a different meaning explanation, and so its summoned concept does not have the same meaning. So long as I follow a precommitment defining a concept, it is hypocritical of me to deny the coherence of that concept, just as it would be hypocritical to deny my language as I speak, to deny my existence so long as I live.
Computation to Canonical Form
We are now free to explore an example of the construction of an ungrounded intuition. I should be specific and point out that not all ungrounded intuitions are under discussion. For the sake of mathematics and logic, intuitions must be completely comprehensible. Unlike grounded intuitions, an ungrounded one may be such that it's never modified by new information. This doesn't describe all ungrounded intuitions, but it describes the ones we're interested in.
One of the most important judgments we will consider is of the form . It is a kind of computational judgment. It's worth explaining why computation is considered before anything else in mathematics. To digress a bit, it’s easy to argue that some notion of computation is necessary for doing even the most basic aspects of ordinary mathematics. Consider, for example, the standard theorem; for all propositions and , . The universal quantification allows us to perform a substitution, getting, for example, , as an instance.
We should meditate on substitution, an essential requirement of even the most basic and ancient aspects of logic. Substitution is an algorithm, a computation which must be performed somehow. In order to realize , we must be doing the activity corresponding to the substitution of with and the action corresponding to the substitution of with at some point. Substitution will appear over and over again in various guises, acting as a central and powerful notion of computation. To emphasize, once substitution is available, we are of the way toward complete and fully general Turing-Complete computation via the lambda calculus. Much of the missing features pertain to explicit variable binding, which we need anyway in order to use the quantifiers of first-order logic. I don't think it's really debatable that computation ontologically precedes logic. One can do logic as an activity, and much of that activity is computational in nature.
Before expositing on some example judgments, we should address the need for isolating concepts. Consider a theory with natural numbers and products . We must ask what constitutes a natural number and a product. By default, we can form a natural number as either zero or the successor of a natural number. e.g. , , , , ... A product can be formed via where is an and is a . Additionally, we have that, if is a natural number then (where is a projection function) is a natural number, and if is a natural number then is a natural number, and if is a natural number then is a natural number, etc. to infinity. This situation gets branchingly more complex as we add new concepts to our theory. If we don't define concepts as fundamentally isolated from each other, we inhibit the extensibility of our logic. This is both unpragmatic and unrealistic, as we will want to extend the breadth of concepts we can deal with as we model more novel things. Furthermore, the coherence of the concept of a natural number should not depend on the coherence of the notion of a product. Ultimately, each concept should be defined by some precommitment consisting of a list of rules for making judgments. If we entertain this infinite regress, then there may be no way in general to state what the precommitment in question even is.
At the core of our definitions will be canonical forms. Every time we define a new concept, we will assert what its canonical forms are. For example, in defining the natural numbers we will judge that and that, assuming , we can conclude that . We can't assume this alone, however. Consider, for example , which should be a natural number, but isn't in the correct form. We now have an opportunity to explain . indicates that we start out with some mental instantiation , and after some mental attention, it becomes the instantiation . So we have, for example . When I say , I do not mean that is equal to . That's a separate kind of judgment. This means our full judgment is that iff or for some . There are some details missing from this definition, but it should serve as a guiding example, the first rough sketch of what I mean by a meaning explanation.
It is worth digressing somewhat to critique the axiomatic method. Most people, especially when first learning of a subject, will experience a mathematical or logical concept as a grounded intuition. This is reflected in a person's answer to questions such as "why is addition commutative?". Most people could not answer. It is not part of the definition of addition or numbers for this property to hold. Rather, this is a property stemming from more sophisticated reasoning involving mathematical induction. A person can, none the less, feel an understanding of mathematical concepts and an acceptance of properties of them without knowledge of their underlying definitions. Axiomatic methods, such as the axioms of ZFC, don't actually define what they are about. Instead, they list properties that their topic must satisfy.
The notion of ZFC-set, in some sense, is grounded by an understanding of the axioms, though it is still technically an ungrounded intuition. This state of affairs holds for any axiomatic system. There is something fundamentally ungrounded about a formal logic, but it's not the concepts which the axioms describe. Rather, what we have in a formal logic is a meaning explanation for the logic itself. That is, the axioms of the logic tell us precisely what constitutes a proof in the logic. In this way, we may formulate a meaning explanation for any formal logic, consisting of judgments for each axiom and rule of inference. Consequently, we can cheaply commit ourselves to the coherence of the logic as an idea. What we can't cheaply commit ourselves to are the ideas expressed within the logic. After all, a formal logic could be inconsistent, it's ideas may be incoherent.
As a consequence, the notion of a coherent idea of ZFC-set cannot be committed to cheaply. This holds similarly for any concept described purely in terms of axioms. It might be made cheap by appealing to a sufficient meaning explanation, but without additional effort, things treated purely axiomatically lack proper definitions in the sense used here.