What is the difference between logic and mathematics?

Question

I’ve read the article in the SEP about the philosophy of mathematics. I believe I follow most of it.

However, I am a bit puzzled by something that may be due to some basic misunderstanding on my part. When it is stated that the goal of (classical) Logicism was to reduce mathematics to logic, what exactly is meant by ‘logic’ beyond the mathematical theory of logic? In what sense, if any, is there a ‘non-mathematical’ logic? We can use natural language to reason logically, but the more concise mathematical formulation is introduced exactly to remove ambiguity when reasoning.

Stated differently: how and why is logic distinct from mathematics in the first place?

Answer 1

Whether there is a distinction, and what the distinction consists in, is a hotly debated topic. Here are a few things that are typically claimed to be essential to logic:

1. Universal applicability: the laws of logic apply to every subject matter. This would mean that, e.g., different theories of arithmetic have the same underlying logic (usually something like classical first-order logic).
2. Ontological neutrality: the thought here is that there are no distinctively logical objects, and nothing need exist for logic to be true. Arithmetic assumes the existence of the natural numbers. Set theory assumes the existence of sets. Logic is supposed to be free of any similar existential assumptions.
3. Epistemic priority: the fundamental truths of logic are in some sense more immune to doubt and more a priori certain than any other subject matter. There are two thoughts here. The first is that knowledge of everything else, including math, requires knowledge of logic. The second is that there is less or no room for doubt when it comes to logic.

Now, each of those claims are debatable. “Universal applicability” seems hard to come by unless you assume a very weak logic, weaker than people typically assume. Classical logic won’t work for intuitionists, and intuitionistic logic won’t capture distinctions central to paraconsistent logics.

Ontological neutrality is similarly debatable. First-order logic is plausibly neutral, but it is relatively weak expressively. For example, it cannot capture the distinction between “finite” and “infinite” and so will be unable to characterize a lot of theories definitely (i.e., you’ll have non-standard models).

Finally, epistemic priority is also up for debate. Depending on what gets included in “logic”, it’s plausible that simple arithmetic and geometric truths are on surer footing or at least more “obvious” than much of logic.

Now, for the early logicists, logic was pretty strong. Russell and Whitehead’s type theory was strong enough to interpret arithmetic — Gödel’s proof of incompleteness is couched in that system — and at least a weak set theory. The problem for them was that Gödel’s results seemed to show that logic couldn’t be the firm foundation they had hoped for, at least not for any mathematics of interest.

The “neo-logicists” make a more modest claim that logic (a weak second-order logic), combined with some conceptual truths about mathematics (like “Hume’s Principle” for the natural numbers), provides a foundation for mathematics. Here the objections will typically be to either the comprehensiveness of the program — it can’t capture all mathematics — or to the logical status of their “logic” (that it’s just “set theory in sheep’s clothing”).

Further reading:

A good survey book on the philosophy of mathematics is Stewart Shapiro's Thinking About Mathematics. Chapter 5 covers logicism and touches on all of these topics, but all of part III (chs. 5 -- 7) is relevant. A freely available discussion of logicism can be found at the SEP. The classic text for neo-logicism (also called "neo-Fregeanism"), championed primarily by Crispin Wright and Bob Hale, is Wright's Frege's Conception of Numbers as Objects. Wright and Hale have a collection of essays on neo-logicism titled The Reason's Proper Study. The SEP entry on "Logic and Ontology" and the entry on "Logical Constants" are also relevant.

More advanced discussion of each of the topics:

1. Universal applicability: this traces back to Kant and especially Frege, who separated (1) and (2) by allowing "concepts" to be part of logic's ontology. Quoting from the SEP entry on "Logical Constants":

…the basic propositions on which arithmetic is based cannot apply merely to a limited area whose peculiarities they express in the way in which the axioms of geometry express the peculiarities of what is spatial; rather, these basic propositions must extend to everything that can be thought. And surely we are justified in ascribing such extremely general propositions to logic. (1885, 95, in Frege 1984; for further discussion, see MacFarlane 2002)

The referenced MacFarlane article is “Frege, Kant, and the Logic in Logicism”. Also see Aldo Antonelli and Robert May's "Frege's New Science" (2000).

2. Ontological neutrality: George Boolos has a good discussion of this in his "On Second-Order Logic" (1975; reprinted in his Logic, Logic, Logic, citations from the reprint). He connects it to (1) under the title of "topic neutrality":

[T]he idea is that the special sciences, such as astronomy, field theory or set theory, have their own special subject matters, such as heavenly bodies, fields, or sets, but that logic is not about any sort of thing in particular, and, therefore, it is no more in the province of logic to make assertions to the effect that sets of such-and-such sorts exist than to make claims about the existence of various types of planets. (p. 44)

3. Epistemic priority: a classic (critical) discussion of this topic is Quine's The Web of Belief. In it he discusses the idea that the truths of logic are somehow more "immune to revision" than other truths. While he is sympathetic to the idea that they are more immune to revision -- more "central to our web of belief" -- he rejects the classic view that they are wholly immune to revision. Chapter 4 on "Self-Evidence" is the most relevant here.

Answer 2

I find that modern students are told the same thing you heard. This was not the case when I learned logic. Mathematical logic did not exist prior to 1845. Notice that I did not say mathematics didn't exist. Aristotelian logic predates mathematical logic and had no symbolization.

Aristotelian logic was semantic. I could say more linguistic. This logic was based on language and context and no symbolization. The art of rhetoric and psychology close are related in how people persaude and deceive other people. Logic that Aristotle expressed was a way to evaluate deceptive reasoning which mathematical logic was not intended for. The way people use words can quickly deceive the weak minded. They speak fast, they use multiple definitions of the same term in the same argument, they use vague terms, and so on. Logical form would allow a listener or observer to quickly recognize deceptive practice. Mathematical logic only cares about validity whereas Aristotelian logic had other rule sets which are lost or renamed.

When I learned logic I was not allowed to use false premises whatsoever. The point being in Logic is to move from truth to other truths which preserves truth on reliability of people using this method of logic. Today people put propositions any kind of way whether true or blatantly false. The mathematicians say logic is about form. This was not always the case. As I said the generation I was taught under did not have false premises allowed. So one would think content mattered to some degree because it Did! How does mathematical logic deal with words in context in a realistic way humans speak. That is what rhetoric does is it not? Think politics here. Smooth talkers who may appeal to emotions ro persuade voters or may flip flop on positions already spoken and answered.

Aristotle had rivals called Sophists who did the same thing as above and which he considered users of the sophitry method bad rhetoric. Thus he wrote a treatise on Rhetoric; and logical treatises were written to make a distinction between the good and the erroneous methods. The way people spoke in reality as far as arguments go was what syllogisms tried to capture. This implies hidden premises and common knowledge claims not stated verbally. How can mathematical logic handle a semantic based system?

The contexts of the premises mattered in Aristotelian logic and not so much symbolized. Even Aristotle divided logic into major logic and minor logic. During the medevial times a term Material Logic emerged which improved upon Aristotelian logic. Shortly after then came symbolic representations. In a famous mathematical conference about the time of 1845 is where the foundation of mathematical logic was laid.

Material Logic likely turned into the field of Epistemology. Material Logic focused on content as well as logical form. Mathematicians don't care about the truth of propositions. Philosophers did. Like I said no false premises were allowed. Thus all arguments needed propositions that were true premises and that necessarily meant there wrere sound arguments. All sound arguments must be valid. That is from philosophy.

Now math does not care about how propositions are formed nor the truth value of the propositions in an argument. Just the form either valid or invalid matters in the mathematical sense. Epistemology today considers what truth values are and what propositions are true not logic alone. Logic when I learned it included firm content and context of arguments. This is not true today.

Answer 3

Whitehead and Russell proved that logic and mathematics are one and the same by deducing mathematics from logical propositions.

Logical propositions are propositions of the form "p implies q." Some propositions are well-know logical propositions, such as "p or p implies p"; some others are dubious as to whether they are logical propositions or not. W & R's primitive propositions are are all well-known logical propositions except perhaps the axiom of reducibility.

See Mathematics and Logic

By the way, if you study logicism long enough, you will find it difficult to put up with people speaking inaccurately. It is a gross misunderstanding to speak of backward extension as reduction. In W&R's words:

Hence the scope of mathematics is enlarged both by the addition of new subjects and by a backward extension into provinces hitherto abandoned to philosophy.

Whitehead & Russell. "Preface." Principia Mathematica. Volume 1. Merchant Books, 1910. v