Mathematics and logicism

Question

If reducing mathematics to logic is the goal of logicism, what other conceptions of mathematics are there?

Answer 1

Philosophy of Mathematics

The philosophy of mathematics has a long and storied history. One of the major questions asked in the field is that of "ontologically speaking, what are the objects studied in mathematics?" Generally, schools of thought in this field can be split into two kinds of theories, mathematical realist theories and mathematical anti-realist theories; further differentiation is made by how the theories formulate those positions.

Realism and Anti-Realism

Mathematical realism asserts that the objects of mathematics (numbers, functions, sets, sheaves, Kälher manifolds, etc.) are in fact ontologically real, abstract entities. Mathematical anti-realism asserts that those objects are not real in the slightest, and are either inherently meaningless results of the manipulation of symbols, "rules of a game", or else that they are entirely made up by humans and have no real existence in the world. An important differentiation that the two types of theories make, which may seem innocuous but is actually very subtle, is whether mathematical theorems/techniques/etc. are discovered or invented.

Platonism

One of the most important views of mathematical realism is mathematical platonism. This view takes its name from Plato and his theory of Forms. Platonism is the view that mathematical objects are real, abstract entities which we experience. This implies that mathematical objects exist independent of humans, in whatever abstract realm abstract objects exist in. From the Stanford Encyclopedia of Philosophy:

Mathematical platonism can be defined as the conjunction of the following three theses:

Existence. There are mathematical objects.

Abstractness. Mathematical objects are abstract.

Independence. Mathematical objects are independent of intelligent agents and their language, thought, and practices.

Importantly, platonism about mathematics often times is at the heart of the debate about universals. The problem of universals is the question as to whether abstract objects such as "love" exist (a realist view) or are things like "love" merely instantiated and exist only as a property of real objects (an anti-realist view). Mathematical objects often times take center stage in these debates because concepts like numbers hold value, conceptually and practically, for our society. One of the strongest arguments for platonism is Putman and Quine's indispensability argument which argues:

(P1) We ought to have ontological commitment to all and only the entities that are indispensable to our best scientific theories.

(P2) Mathematical entities are indispensable to our best scientific theories.

(C) We ought to have ontological commitment to mathematical entities.

It's important to note that logic isn't needed for platonism to work. Platonism is the view that mathematical objects exist and their structures are all matters of fact about the objects themselves, not about a conceptual framework we can use to describe them. Even further, it might be true that there are mathematical truths that are not reducible to a logical system. If that were the case, platonism would be fine but logicism would not.

Nominalism

As a converse to mathematical platonism, there is also the view of mathematical nominalism which is the corresponding anti-realist view in the problem of universals outlined above. Nominalism's main concern is the denial of the platonist view that mathematical objects are abstract objects. Arguments such as Quine's are heavy enough to require responses from nominalists so much of the contemporary work done in this school of thought is to save science (as well as regular mathematics) from platonism, so to speak:

Broadly speaking, there are two forms of mathematical nominalism: those views that require the reformulation of mathematical (or scientific) theories in order to avoid the commitment to mathematical objects (e.g., Field 1980; Hellman 1989), and those views that do not reformulate mathematical or scientific theories and offer instead an account of how no commitment to mathematical objects is involved when these theories are used (e.g., Azzouni 2004).

A nominalist would argue that if you are holding two apples, then the concept of "two" is a coherent thought; however, they would argue that the abstract idea of "two" is not embodied by any sort of abstract entity that exists. It is the same idea as was outlined above with the problem of universals. A nominalist doesn't deny that "love" is coherent concept, they merely deny that "love" is an abstract entity that exists independently of things that instantiate it. Mathematical nominalism has the same view of mathematical objects. Nominalism also is not committed to logicism by definition. Nothing about nominalism, the rejection of abstract mathematical entities, entails that mathematics must therefore be reducible to logic. The former could be true and there could still be mathematical theorems that are not expressible in a logical system.

Formalism

A very important theory is that of formalism, historically championed by David Hilbert. Formalism is an anti-realist view that says mathematics, and logic, are nothing more than manipulation of symbols. Formalism places the emphasis of its argument on the assertion that mathematics is just a game and that rules of deduction are merely rules explaining how to manipulate symbols. Hilbert's axiomization of geometry was the impetus for his own formalist views. In Foundations of Geometry he creates a system of axioms and rules of inference that behave in a specific, desirable way. After outlining this system, Hilbert then explains that if we interpret the objects of the system as "lines" and "points" and so on, we are left with a system that correctly describes Euclidean geometry. This idea that the interpretation of the theory is independent of what the theory says is summarized as follows:

Hilbert's central idea, again, is to focus not on particular geometrical concepts like point and line, but to pay attention instead to the logical relations that are said, by the axioms, to hold between those concepts. The question of whether the parallels axiom is independent of the other Euclidean axioms has entirely to do with the logical structure exhibited by these axioms, and nothing to do with whether it is geometric points and lines one is talking about, or some other subject-matter altogether.

Formalism, as studied today, is slightly different in content than Hilbert's, as is outlined in the SEP article. However, at its core formalism is the view that mathematical and logical content is nothing more than the effects of specific syntactical rules, i.e. string manipulation. As such, formalism is considered to be a form of mathematical anti-realism.

Logicism

Formalism can be, for conceptual and historical reasons, compared and contrasted with logicism. Logicism is the philosophical view that mathematics can be reduced to logic, and therefore mathematics is merely just a larger part of logic as a whole. Historically, logicism was defending and elaborated on by Frege, Dedekind, Peano, and Russell, among others. Frege's goal for logicism was to create a purely logical system that could provide all of the truths about arithmetic. He failed to do so, as his system was subject to Russell's paradox.

Russell attempted himself to create a system that avoided this and a few other paradoxes, his system of Principia Mathematica which laid the foundation for type theory. Kurt Gödel's metamathematical results about PM (and all other logical systems capable of Robinson arithmetic), his incompleteness theorems, seemed to many to be a final blow to logicism, in that no single logical system will ever be able to encompass all truths of mathematics and at the same time prove that itself is consistent. Some modern interest has been paid to a revival of logicism, called neo-logism, due to the work of Crispin Wright showing that Frege's original plan might be salvageable by replacing his Basic Law V (the cause of the failure of his system, besides the failure Gödel's results would imply) with Hume's Principle. Current criticism of neo-logicism is that it still fails to address some philosophical issues raised by logicism such as the Julius Caesar problem.

Logicism may be neutral on the ontological status of mathematical objects. What is important to note is that logicists agree that logical truths are just that, truths. More specifically, they agree that logical truths are analytic truths, truths that are guaranteed to be true by the very nature of their definitions, and are knowable a priori, knowable without direct experiences.

An important note is the disagreements Frege and Hilbert had about the role logic and axioms play in mathematics. This debate was centered around the proper role that axioms play in logical systems and how we should interpret them. As such, this debate exemplifies many of the tenants that contrast formalism with logicism. From the SEP:

Because the content of the geometric terms is irrelevant to the issues of consistency and independence with which Hilbert is concerned, it is immaterial from his point of view whether one understands his axioms to be (a) fully-interpreted sentences whose geometric terms have their ordinary geometric meanings, (b) fully-interpreted sentences whose geometric terms take on one of Hilbert's re-interpretations, or (c) partially-interpreted sentences whose geometric terms appear simply as place-holders.

For Frege on the other hand, the differences just listed are crucial, particularly since the consistency of the thoughts expressed by the sentences construed as in (a) above is not implied either by the consistency of the thoughts expressed by those sentences understood as in (b) or by the satisfiability of the relation defined by the sentences understood as in (c). Hence we can see both why Frege found Hilbert's cavalier attitude regarding the distinctions between (a), (b), and (c) to be virtually incomprehensible, and why Hilbert found Frege's criticisms entirely unconvincing.

Frege and Russell believed to their core that mathematics was about mathematical objects: that mathematical, and therefore logical, propositions spoke about things. Their reduction of mathematics to logic was not their concession that mathematics is just syntax and meaningless symbol manipulation. To that extent, they believed that objects of mathematics and logic are fundamental principles that exist independently of any games we decide to write down. This is the view that contrasts formalism and logicism and is an example of realism.

Intuitionism, Constructivism, Finitism, and Ultrafinitism

These next schools of thought are all closely related in that they reject certain aspects of mathematics and logic (however they are not all the same thing). Intuitionism is the view, first proposed by L.E.J. Brouwer, that mathematics is purely a mental activity. In this respect it is somewhat similar to idealism, however it is not the same thing as Kant's philosophy of mathematics. From the SEP:

Intuitionism is based on the idea that mathematics is a creation of the mind. The truth of a mathematical statement can only be conceived via a mental construction that proves it to be true, and the communication between mathematicians only serves as a means to create the same mental process in different minds.

Constructivism says that the only mathematical objects that exist are those that can be constructed and therefore it rejects proof by contradiction, or any other proof that does not explicitly construct an object. Finitism is an example of constructivism and it maintains that the only mathematical objects that exist are those that can be constructed but purely finite means. This means that a finitist would agree that there could be some arbitrary large natural number N, which is constructed by applying the successor function to 0 N times, for example. However, a finitist would reject any infinite number, such as an infinite ordinal or cardinal, because they are unable to be constructed via finite means. Ultrafinitism is an even more extreme form of finitism which says that finitism should also be restricted to only constructive processes that are plausible in practice. They reject construction of numbers that are arbitrarily large and say the only numbers that exist are those that could feasible be created given the physical limitations of the universe. There are estimations of the number of sub atomic particles in the observable universe but this number is already too large to construct in a practical way, so some ultrafinitists would reject its existence.

The main connection between intuitionism and constructivism is that both of these systems reject mathematical objects that cannot be constructed. As such, they both reject the law of the excluded middle (the law that says, for every proposition P, either P or ~P is true). Intuitionistic logic is a formal logic that follows the ontological views of constructivism and intuitionism. Due to this fact, most theory done in these schools of thought are done in intuitionistic logic. From the SEP:

Philosophically, intuitionism differs from logicism by treating logic as a part of mathematics rather than as the foundation of mathematics; from finitism by allowing constructive reasoning about uncountable structures (e.g. monotone bar induction on the tree of potentially infinite sequences of natural numbers); and from platonism by viewing mathematical objects as mental constructs with no independent ideal existence.

These are all examples of anti-realism.

Conclusion

So, after laying out many of the schools of thought in the philosophy of mathematics, the easiest way to show how logicism differs from other branches is to stress logicism's commitment to, well, logic. Logicism says that mathematics is reducible to logic, above and beyond anything else. Other schools of thought are not required to make this commitment. Platonism can say that "Look, most of mathematics can be expressed in a logical system, but due to results of Gödel and Tarski there will never be one logical system that can encompass all truths about the natural numbers or define truth within itself. However, those truths are about numbers and numbers exist, therefore those truths do exist even though they cannot be reduced to a logical system." Interestingly, Gödel defended platonism.

Similarly, formalism only has commitment to the games that are being played. A formalist can say "Look, we aren't talking about circles and lines and functions, we are just talking about symbols and rules that explain how to move those symbols around. Gödel's results are a deep and profound insight into the actual properties of these systems, and those properties exist no matter what sort of interpretation we give to them. If a system cannot ever tell the whole story, and that is a meta-property of the system, then that is the way mathematics is." They do not have to have commitment to logic being able to provide a complete view of mathematics.

Logicism itself cannot shrug off these results quite as easily. Gödel's incompleteness theorems have a tendency to be used in completely unwarranted ways and to say things that they do not say; however, one thing that they do say is that the enterprise of logicism is severely wounded. Again, logicism's main commitment, the commitment that defines it, is that mathematics is reducible to logic. Sure, a very large part of mathematics is reducible to purely logic, but Frege and Russell's original formulations of logicism wasn't the idea that only some of mathematics is reducable to logic, it was that mathematics itself is just a large part of logic.

A crude, and someone reductive way to answer the question of "besides reduction to logic, what other views of mathematics are there," would be to say: mathematical entities are real, abstract objects regardless of whether or not logic can describe them; mathematical objects do not exist and we're just playing games with syntax; or mathematics is purely a mental activity that does not exist unless an intelligent being is there to intuit them, regardless of whether or not a logical system can describe them This is only to name a few.

For further reading see any of the references used, which are mostly introductions or short surveys, and the references therein.

Answer 2

Well, there is a midway point in the programme of logicism; which is reducing mathematics not to logic but to set theory.

Now, in this theory it is the notion of a set that is basic, and then the notion of a function is derived; it turns out that one can reverse this, and take the notion of a function as basic, and the notion of a set as derived; this theory is called category theory, and in fact if you axiomatise the notion of a function this is what you get; it has more of an algebraic flavour than set theory, which has more of a logical flavour; which perhaps isn't surprising, when one considers their background - set theory came out of investigations of logic, and category theory from algebraic topology.

Category theory didn't come out of study of foundations; in fact, the authors of the theory, in their first penned paper on it, supposed it would be the last!

It's only in the last few decades that Category Theory has been mooted as an alternative foundation for mathematics; this will be an uphill struggle, given how much effort has been invested already in Set Theoretical foundations.

Category theory tends to be plural - there are many useful categories, of which Set is one; whereas Set Theory insists that Set is the one and true category of which all others bud off; personally speaking, I prefer the plural vision.