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20th Century Analytic Philosophy

What is Logicism?

Logicism can be defined as, “the thesis that mathematics is reducible to logic, hence

nothing but a part of logic.” (Carnap, pp. 41) Furthermore, the logicist thesis can be split into two

distinct parts: “1) The concepts of mathematics can be derived from logical concepts through

explicit definitions. 2) The theorems of mathematics can be derived from logical axioms through

purely logical deduction.” (Carnap, pp. 41) One might reasonably ask what benefits are to be

gained by pursuing a logicist agenda. As Gottlob Frege said, “Language proves to be deficient,

however, when it comes to protecting thought from error. It does not even meet the first

requirement which we must place upon it in this respect; namely, being unambiguous.” (Frege,

pp 84) In ordinary language, the same word can sometimes have more than one meaning, and it

is this ambiguity that leads to confusion and imprecision. Consider the sentence “Marzipan is

almond paste and sugar.” with “Marzipan is more expensive than licorice.” In both of these

sentences, the word ‘is’ is used in an ambiguous fashion; the first sentence uses ‘is’ to explain

what marzipan is constituted of. In the second sentence, ‘is’ is used to denote a two-place

relationship between marzipan and licorice. A purely symbolic and logical language would do

away with such ambiguity, as it is the purpose of the symbols and logical operators to make

explicit the precise meaning of a statement. Although mathematicians use a great deal of

symbolic notation in their proofs, they still rely on ordinary language to communicate their ideas.

Thus, it would provide additional clarity to the mathematician if they could somehow translate

all of their proofs into purely symbolic terms.

Logicism draws its inspiration from an observation made by many mathematicians prior

to and including Frege. Namely, “Mathematicians in their investigations of the interdependence

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of mathematical concepts had shown … that all the concepts of arithmetic are reducible to the

natural numbers.” (Carnap, pp. 42) This is simple enough to deduce: from an assumption of the

existence of naturals one can construct the integers by defining the concept of a negative

numbers, and from the integers deduce the rationals by explicitly defining a group operation, etc

all the way to the field of complex numbers. Most mathematicians simply assumed the natural

numbers to exist, because they are such an intuitive concept that even a child without an

education in mathematics makes use of the counting numbers. But the logicists, in their quest for

absolute certainty and radical minimalism, decided to question even the seemingly obvious

notion of natural numbers. Thus, it was the logicists who were the first to not only ask, “Can the

set of natural numbers be deduced from an even more simple set of axioms?”, but to also attempt

proving that the answer is a resounding “Yes!” It is in this context that logicism has developed as

a philosophy of mathematics.

Another motivation of the logicist school was to show how logic could be used as the

foundation of all the theorems of mathematic, and in this respect function as a unifying thread

through which the science of mathematics is woven. This is one of the main goals of the logicist

school, and served as the motivation behind the Principia Mathematica, a scholarly collaboration

primarily between Alfred North Whitehead and Bertrand Russell to derive all of mathematics,

both its theorems and the concept of numbers, from a select few axioms of logic. This had been

tried many times in the past, most notably by Gottlob Frege in his Grundgesetze and

Begriffsschrift. However, the discovery of Russell’s Paradox contradicted Basic Law V, one of

the foundational axioms from which Frege built his language in the Grundgesetze. Even though

the paradox served as a fatal blow to Frege’s particular attempt, Russell was still determined and

convinced that it was possible to deduce all of mathematics from logic alone. And thus, he and

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Whitehead set out to write the Principia. By avoiding Frege’s notion of types, which rely on

assuming that a function can use itself as an argument, and instead relying on the idea of a

‘hierarchy of classes’, Russell thought he had finally discovered the secrets to unlocking the

logical foundations of mathematics while avoiding the use of self-referential formula. However,

“Although Principia succeeded in providing detailed derivations of many major theorems in set

theory, finite and transfinite arithmetic, and elementary measure theory, two axioms in particular

were arguably non-logical in character: the axiom of infinity and the axiom of reducibility.”

(Irvine, Principia Mathematica)

Although the Principia was ultimately unsuccessful in carrying out the logicist program,

it still had a profound historical impact on the history of analytic philosophy. In particular, the

Principia directly influenced the works of Ludwig Wittgenstein, in specific his Tractatus

Logico-Philosophicus. In turn, it was Wittgenstein’s Tractatus that gave the Vienna Circle their

original purpose: to meet together in order to discuss and analyze passages from the Tractatus

line by line. And it was from the ideas expressed in the Tractatus that the Vienna Circle

developed their philosophy of logical positivism.

The logical atomism present in the Tractatus greatly influenced the positivist idea of the

Verification Principle. By dividing the world into atomic facts with definitive true-false values,

Wittgenstein gave rise to the verificationism of A.J. Ayer, the falsificationism of Karl Popper,

and the language-analytic methods of the Vienna Circle and logical positivism. This idea that

one could translate the world into purely logical terms draws its inspiration from the Principia

and its attempt to translate all of mathematics into purely logical terms. One could go so far as to

say that logical positivism is an intellectual cousin to logicism. The Principia sought to formulate

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mathematics in purely symbolic terms, and this reductionist technique of analysis is ever present

in the works of logical positivists.

Logical positivism was also influenced by logicism in its treatment of mathematics as an

analytic statement. For the logical positivists, a large part of their epistemological theory relied

on the analytic/synthetic division. According to the logical positivists, all statements fell into the

category of analytic or synthetic. An analytic statement is a statement that is true by virtue of its

definition, and synthetic statements are defined as all those statements which are not analytic. In

other words, “the [logical] positivists held logic and mathematics to be a priori and denied that

they contained informative truths.” (Skorpuski, pp. 52) To the logical positivists, logic was

vacuous and tautological. Logic says nothing about the particular content of a statement, as logic

is only concerned with the general form. Since logic lacks content, it is impossible to contain any

sort of informative truth, according to the positivists. This belief stems from Wittgenstein in the

Tractatus, who writes, “The propositions of logic are tautologies. | The propositions of logic

therefore say nothing. (They are the analytical propositions.)” (Wittgenstein, 6.1 & 6.11)

Wittgenstein made it clear that logical statements are in this sense devoid of content, but it was

the work of logicism to spread the idea that mathematics is nothing more than an extension of

logic. Thus, it is appropriate to credit logicists for the Vienna Circle treating not just logical

statement, but also mathematical statement, as analytic a priori.

Clearly logicism had an impact on the development of logical positivism, which in turn

became one of the most influential schools of thought in 20th century philosophy. But the

question still remains as to whether logicism is still plausible as a metaphysical theory of

mathematics. I believe that although the original aim of logicism is not an attainable goal in the

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wake of Gödel’s Incompleteness Theorems, logicism could be reformed to become a working

foundation with slight modifications to its overall aims.

Much research in recent years has revolved around so-called neologicism, which is a

philosophical attempt to revive and update the logicist school to be more in line with the

developments of formal logic and mathematics. More specifically, neologicism still seeks to

maintain the original spirit of logicism by bringing mathematics under the scope of logic, but it

must somehow become less restrictive in its claim, lest it fall victim to the same traps Russell’s

and Frege’s theories fell victim to. In specific, the original goal of logicism can be accomplished

by neologicism through one of three methods: “1) Expand the conception of what counts as

‘logic’. 2) Allow more resources than ‘logic alone’. 3) Reconceive the notion of ‘reducible’.”

(Bernard and Zalta, pp. 6)

Although all three reformations of logicism have their merits, I believe that the third

approach, to reconceive the notion of ‘reducible’, yields the most plausible and ontologically

sound results, while still closely maintaining the original spirit of logicism. In particular:

Our view is that philosophy itself should not be concerned with ‘mathematical’ foundations for mathematics. … Philosophers should be concerned with metaphysical and epistemological foundations for mathematics, and we therefore plan to offer a notion of reduction that provides answers to the metaphysical question, “What is mathematics about?”, and to the epistemological question, “How do we know its claims are true?” Indeed, a unique feature of our program is that it yields no proper mathematics on its own, and so makes no judgments about which parts of mathematics are philosophically justified! Instead, it takes as data any arbitrary mathematical theory that mathematicians may formulate, and provides a more general explanation and analysis of the subject as a whole. This analysis encounters no limits of abstraction. (Bernard and Zalta, pp. 35)

*The principle κτ = ıx(A!x & ∀F(xF ≡ τ |= Fκτ)), “which guarantees that the object κ of theory τ is the abstract individual x which encodes all and only the properties that κ has in τ.” (Bernard and Zalta, pp. 36)

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In order to accomplish this bold goal, a sufficiently powerful and expressive logic must be used.

Bernard and Zalta opt to use third-order non-modal object theory as their language of choice.

The reasoning for choosing 3rd order object theory is because, “Whereas this principle* for

identifying the mathematical objects of the theory τ employs only second-order object theory,

we need third-order object theory to similarly identify the properties and relations of τ.”

(Bernard and Zalta, pp. 36)

Bernard and Zalta’s neologicism has many strengths which other conceptions lack. The

first advantage it has is that their theory requires only one abstraction principle for converting all

mathematical objects to logical constructs, regardless of their original domains and the particular

axiomatization. This makes it minimalistic, uniform and elegant, as opposed to other forms of

neologicism which have a separate abstraction principle for each particular axiomatization. It is

also worth mentioning that their logic is non-modal. This is a nice property because modal logic

is often used by neologicists to skirt around the issue of implicitly assuming some sort of

mathematical object by relying on the possibility operator. By stating “It is possible that…” a

modal system of logic avoids asserting the existence of abstracta by merely considering the

possibility that they exist. Although this technically works to some degree, it isn’t a very

satisfactory solution because the overall goal of reducing math to logic becomes bogged down in

the details of formulating lengthy and complex modal statements. Furthermore, reducing

mathematical abstracta to a possibility is still not a satisfactory answer. To many, this comes off

as a cheap way of avoiding ontological questions. Thus, it is in this respect that one must give

credit to Bernard and Zalta for formulating their neologicism without relying on modal logic; it

greatly simplifies things and shows their commitment to genuinely answering philosophical

questions.

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The second advantage is that Bernard and Zalta’s neologicism satisfactorily addresses

epistemic questions regarding mathematical knowledge. Specifically, their theory answers the

concern “How do we know the claims of mathematics are true?” By using their third-order

reduction formula, all the theorems and axioms of mathematics can be converted into the

language of third-order object theory. Because expressions of formal logic are

necessarily analytic, we can thus state with certainty the truth-value of any particular

mathematical claim.

Finally, Bernard and Zalta’s neologicism gives a firm answer to the other genuine

philosophical question about mathematics: “What is mathematics about?” In their own words,

“Our answer is that mathematics is about abstract objects (indeed, objects that bears some

resemblance to the ‘indeterminate elements’ [Benaceraf 1965] required by structuralist analyses

of mathematics) and the properties that they encode.” (Bernard and Zalta, 39) This answer is

particularly appealing because it is general enough to allow wiggle room for the expert

mathematicians to come along and decide precisely what those particular abstract objects are, but

still specific and concrete enough as to provide a satisfactory answer to the curious philosopher.

In summary, logicism is an extremely important school of thought that had a tremendous

influence on 20th century analytic philosophers, in particular the Vienna Circle. The logicist goal

of reducing all of mathematics to logic was a broad one that shined great insights on the nature of

mathematics, gave rise to the most significant development in logic since Aristotle, and paved

the way for the Vienna Circle and the logical positivism that dominated philosophical thought in

the 20th century. Of particular influence was the Principia Mathematica, a book published by

Russell and Whitehead that attempted to reduce mathematics to a purely logical schemata of “the

hierarchy of types”. Although this particular effort was not successful, it inspired Ludwig

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Wittgenstein and his logical atomism, which in turn directly influenced the Vienna Circle.

Although the attempts of Frege and Russell were too ambitious in their goals (a fact eventually

proven by Kurt Gödel), their efforts were not in vain. Their efforts inspired the school of

neologicism, which is a modern attempt to revive the logicists program of showing mathematics

can be reduced to logic. Although neologicists have to tone down their goal in order to be

consistent with the discoveries of Gödel and other developments of logic, the spirit of

neologicism nonetheless remains loyal to the overall goals of the original logicists, and I am sure

that if Frege or Russell were alive today, they would enthusiastically support the neologicist

revival of the program those two men cherished so deeply.

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Works Cited

Carnap, Rudolf. "The Logicist Foundations of Mathematics." Philosophy of Mathematics: Selected

Readings. 2nd ed. New York City: Cambridge University Press, 1983. Print.

Frege, Gottlob. "On the Scientific Justification of a Conceptual Notation." Conceptual Notation & Related

Articles. pp 83-89.

Irvine, A. D., "Principia Mathematica", The Stanford Encyclopedia of Philosophy (Winter 2010 Edition),

Edward N. Zalta (ed.), URL = <http://plato.stanford.edu/archives/win2010/entries/principia-

mathematica/>.

Irvine, A. D., "Russell's Paradox", The Stanford Encyclopedia of Philosophy (Summer 2009 Edition),

Edward N. Zalta (ed.), URL = <http://plato.stanford.edu/archives/sum2009/entries/russell-

paradox/>.

Linsky, Bernard, and Edward N Zalta. "What is Neologicism?." Bulletin of Symbolic Logic. 12.1 (2006): 60-

99. Web. 21 Sep. 2011. <http://mally.stanford.edu/Papers/neologicism2.pdf>.

Skorpuski, John. "Later Empiricism and Logical Positivism." The Oxford Handbook of Philosophy of

Mathematics and Logic. 1st Ed. New York, New York: Oxford University Press, Inc., 2007. Print.

Wittgenstein, Ludwig. Tractatus Logico-Philosophicus. New York City: HarperCollins Publishing, 2009.

Print.