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Brandon Fitzpatrick

Dr. Hasanoglu

Philosophy of Mathematics

May 6, 2019

A Neologicist’s Critique of Curry’s Term Formalism

It appears on the face of things that we come to know mathematical truths differently

from scientific truths. People generally agree that scientific truths are known causally. That is to

say that, one knows that objects fall down because one sees objects fall to the ground.

Mathematics truths are not “seen,” they are not known through empirical observation. The

challenge is to create an epistemological theory for mathematical statements and then determine

their ontological consequences. Two major views address this challenge, logicism and

formalism. Logicism, first put forward by Gottlob Frege, argues that mathematics is reducible to

logic. Formalism, on the other hand, argues that mathematics is mostly symbol manipulation

according to specific rules. I argue that Curry’s formalism does not avoid the problems of older

term formalism and Hilbert’s program, instead we should opt for the neo-fregean logicist view.

Gottlob Frege, a mathematician and logicist, was one of the first to address the

epistemological challenge and began the school of logicism. The main argument of logicism is

that mathematics is reducible to logic. Mathematical truths are in the end logical truths.

For Frege, mathematics is analytic. But Frege redefines Kant’s notion of analytic truth.

In the beginning of the Critique of Pure Reason Immanuel Kant defines the distinction between

analytic and synthetic truths as, “either the predicate B belongs to the subject A, as somewhat

which is contained (though covertly) in the conception A; or the predicate B lies completely out

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of the conception A, although it stands in connection with it. In the first instance, I term the

judgement analytical, in the second, synthetical,” (Kant 31). The idea of containment is to mean

if one concept A contains another concept B then to understand A one necessarily understands B.

For example, the statement, “all bachelors are unmarried” is analytic because contained in the

concept bachelor is the concept unmarried. The statement, “all bachelors are unhappy,” is

synthetic because the concept bachelor does not include unhappy. According to Kant, an

analytic statement is a statement whose predicate is contained within the subject.

Due to Kant’s understanding of the analytic synthetic distinction, Kant argued that

mathematics are synthetic a priori truths. Kant’s reasoning is as follows; in the statement

7+5=12, 12 does not contain the idea 7+5. To understand 12 does not necessitate the

understanding of 7+5. These mathematical statements must be synthetic due to their lacking of

subject-predicate containment. Since these statements aren’t known through sense experience,

they are a priori as well. Kant claims that mathematical statements are synthetic a priori

statements.

Frege redefines analyticity through definitions. He was concerned with how we come to

know these statements. The definition of a statement is grounded, it is of the world. These

analytic statements are then not trivial but telling you something about the world. But these

statements are a priori for Frege, they aren’t contingent on sense experience to be known.

Analytic truths are true by their definitions and by logic. Relating back to mathematics:

if [the statement] concerns a mathematical truth. It now depends on finding a

proof and following it back to the primitive truths. If, on the way, only general

logical laws and definitions are encountered, then the truth is analytic, [...] and if,

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on the other hand, it is possible to provide a proof from completely general laws,

which themselves neither need nor admit of proof, then the truth is a priori. (Frege

93)

Mathematical truths are analytic and a priori, rather than the Kantian idea of mathematics being

synthetic a priori. We prove mathematical statements using the general logical laws and

definitions.

Now to definitively show that mathematics is reducible to logic, Frege must show how

one derives mathematics from logic. Frege begins with deriving arithmetic for only Hume’s

principle. Hale and Wright define the principle as “the number of Fs is equal to the number of

Gs if and only if there exists a one-to-one correspondence between the Fs and the Gs,” (Hale

Wright 22). The amount of things under two concepts are equinumerous if there is a bijection

between their elements. Using only this principle, Frege was able to derive the Peano axioms for

arithmetic.

It’s important to note that Frege did not consider Hume’s principle a logical law. Nor did

he think that this principle was able to help us come to know about numbers, since the principle

was unable to define what a number was. Frege thought this about Hume’s principle due to what

is referred to as the Julius Caesar problem. Frege argues that using Hume’s principle as a

definition of numbers:

we can never [...] decide by means of our definitions whether the number Julius

Caesar belongs to a concept, or whether that well-known conqueror of Gaul is a

number or not. Furthermore, we cannot prove with the help of our attempted

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definitions that if the number a belongs to the concept F and the number b belongs

to the same concept, then necessarily a= b. (Frege 106)

Hume’s principle does not necessarily make it the case that Julius Caesar is not a number, nor

that two numbers belonging to the same concept are in fact the same number. In order to avoid

this problem Frege relied on Basic Law V. Frege used Basic Law V to derive Hume’s principle

in such a way as to avoid the Julius Caesar problem and to have the foundation of mathematics

be a logical law.

Basic Law V is what Frege believed was a logical law. Law V is about the extensions of

concepts. The extension of a concept is the set of things of which the concept applies to. Zalta

defines Law V in the Stanford Encyclopedia of Philosophy as, “the extension of the concept F is

identical to the extension of the concept G if and only if all and only the objects that fall under F

fall under G,” (Zalta). For Frege, the extension of the concept of a particular number is actually

comprised of sets. For example, the extension of the concept three is all three numbered sets.

Redefining Basic Law V in terms of sets, we get the definition the set of Fs is equal to the set of

Gs if and only if the F and G are materially equivalent or share the same true value. Using Basic

Law V Frege proves Hume’s principle.

Briefly sketched is Frege’s proof of Hume’s principle from Law V. For the first direction

assume that the number of Fs is equal to the number of Gs. The extension of equinumerous to F

and the extension of equinumerous to G are equal. We know F is equinumerous to F so by

substituting we get the extension equinumerous to F is equinumerous to F. Then substituting

again we get the extension equinumerous to G is equinumerous to F. Then in the end we get G is

equinumerous to F. For the other direction, we assume F is equinumerous to G and we want to

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show the number of Fs is equal to the number of Gs. By extensions, the extension equinumerous

to F is equal to the extension equinumerous to G. Then to show F and G are equinumerous, one

needs only to show that their extensions are the same.

Now Frege has shown arithmetic can be reduced to Hume’s principle and Hume’s

principle can be reduced to Basic Law V. Though Frege did not prove this, he believed all

mathematics beyond arithmetic could be reduced this way as well.

Bertrand Russell, another logicist and mathematician, showed a contradiction in Frege’s

Basic Law V. Russell showed that Frege’s law of extension implies the existence of a

paradoxical set, that poses problems for Frege’s program and naive set theory. Consider the set

A, set X is an element of A if and only if X is not an element of X. That is A consistent all and

only the sets that do not contain themselves. The question is if A is an element of A. A by its

own definition must contain itself if and only if it does not contain itself. Basic Law V allows

for this paradoxical set to exist. A less technical example Bertrand Russell uses to illustrate this

paradox is the case of the barber. The barber shaves all and only the men who do not shave

themselves. Does this barber shave himself? If he does shave himself then he does not shave

himself because the barber shaves all and only the men who shave themselves. If he does not

shave himself then he shaves himself because the barber shaves all and only the men who do not

shave themselves. And so this barber is a paradox. If we accept Basic Law V then there exists a

paradoxical set.

With this contradiction Frege’s derivation of Hume’s principle no longer holds, since

Basic Law V cannot be a logical law. Now Hume’s principle is no longer reducible to logic nor

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the rest of arithmetic in the manner in which Frege wanted. After Russell showed this flaw in his

proof, Frege abandoned his program.

Years later Bob Hale and Crispin Wright attempt to salvage Frege’s program, forming a

school of neo-fregeanism. They point to the fact that Frege’s derivation of the Peano axioms

from Hume’s principle is considered sound. Though the derivation of Hume’s principle from

Basic Law V is faulty, they show that Frege only used Basic Law V in that derivation, and

besides that, it is not used in his derivation of the Peano axioms. They argue, then, that we

should not consider Basic Law V at all, and simply begin with Hume’s principle.

Using Hume's principle, they argue that we can abstract numbers from it. “A statement

of numerical identity—in the fundamental case, a statement of the kind: the number of Fs = the

number of Gs—is true in virtue of the very same state of affairs which ensures the truth of the

matching statement of one-one correspondence among concepts, and may be known a priori if

the latter may be so known” (Hale Wright 25). Hume’s principle for Hale and Wright is an

example of an abstraction principle.

An abstraction principle is a statement were a new thing can be discovered from

understanding the statement. This is similar to the relationship between types and tokens. If one

has several objects or tokens of different properties except all share the property of being red,

one can abstract the new thing or type of redness. Redness is not found anywhere but only

through an abstraction from several tokens, it can be discovered.

In the case of Hume’s principle the abstraction is slightly different. Rather than seeing

several things and abstracting, one must understand the statements and can thereby abstract. If

one understands the statement the number of Fs is equal to the number of Gs if and only if the Fs

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and Gs are equinumerous, then one can discover the existence of this abstract object called

numbers. Types and tokens rely on a posteriori knowledge, Hume’s principle relies on a priori

knowledge. The process of discovering types is only through sense many instances of tokens

and so it is empirical and a posteriori. Abstraction of Hume’s principle does not really on senses,

one does not to see many sets to make this abstraction. So the abstraction is through

understanding rather than empirical investigation. The understanding of Hume’s principle is

analytic. To learn of numbers one only needs to understand that the truth of one side of Hume’s

principle necessitates the truth of the other.

Another school of the philosophy of mathematics is formalism, whose most prominent

proponent is David Hilbert. Logicism was very concerned with the semantics of mathematics,

asking: What do these mathematical statements mean? But the formalist turns their attention to

the syntax of mathematics. The formalist is less concerned with the meaning of these symbols

and much more concerned with the rules by which they are governed.

Term formalism is about the naming mathematical objects. The number two only means

the symbol “two”. A number is the same as its name. Mathematical objects are merely their

symbol. Knowledge of mathematics is knowing the relationships between the symbols.

Mathematics is merely a certain collection of symbols.

But a problem arises for the term formalist: How does one make a term for every

number? The natural numbers are easy to name because they are countable. And constructing

the integers are easy, simply add in the inverses of the natural numbers, for example, the inverse

of 2 is -2 , adding those to the list of natural numbers, so you would have 0, 1, -1, 2, -2, and so

on. Even rational numbers are possible because their decimals have a terminating digit. But the

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real numbers are continuously infinite. Consider the case of the number e defined as the base of

the natural log. The number e is between 2 and 3, but even more precisely between 2.7 and 2.8,

but even more so between 2.71 and 2.72, and this process can continue to infinity. It seems

impossible to make a term for every real number.

Game formalism is similar to term formalism but is more concerned with the syntax. In

game formalism mathematics is like a game. Before you start everyone agrees upon the rules

and the objects to be used. Then it’s just a matter of symbol manipulation. Mathematics is not

about anything, there are no semantics. An earlier game formalist Thomae, was quoted by Frege

in Frege’s Basic Laws of Arithmetic as describing game formalism as:

arithmetic is a game with signs which one may well call empty, thereby

conveying that (in the calculating game) they do not have any content except that

which is attributed to them with respect to their behaviour under certain

combinatorial rules (game rules). A chess player makes use of his pieces in a

similar fashion. (Frege 88)

One can give these symbols meaning but those meanings are not necessary to practice

mathematics. Mathematics is empty symbol manipulation according to syntactic rules.

Similarly the game formalist attempts to sidestep the metaphysical and epistemological problems

associated with mathematics. The objects of mathematics are the symbols we choose to use

without any meaning outside of their relationship, according to the syntactical rules, to other

symbols. We come to know mathematics through understanding the rules of the game.

But this version of formalism arguably fails due to Frege’s critique of its inability to

explain the applicability of mathematics. If mathematics is merely a game, then why is it so

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useful in the sciences? Consider the game of chess. The pieces hold no semantic meaning and

the rules are arbitrary. The game is played following the rules with the pieces, and comes to the

result of winning, losing, or drawing. The game of chess is not applied in any other field the

same way mathematics is. The products from the “game” of mathematics seem to have genuine

usefulness in the sciences. If mathematics is merely symbol manipulation according to syntactic

rules then it would seem there should not be such a strong applicability to the sciences. This

applicability seems to be explained by mathematics being more than mere symbol manipulation.

Hilbert attempts to solve the shortcomings of these more naive formalist theories. Hilbert

was concerned not with the truth of mathematics but with the consistency of it. In the year 1900

Hilbert made a list of the most important problems of mathematics to be solved in the next

century. The second question was to prove the axioms of arithmetic to be consistent. And so

Hilbert’s program begins.

Hilbert’s formalism relies on deductivism. A statement is true if it is deducible from a

given set of axioms. On the face of it this view is similar to Frege’s logicism, but the difference

is the standing of the axioms. Frege argued that the axioms are logical laws. Hilbert on the other

hand argued that they are non-logic and could be considered arbitrary. The axioms can be given

meaning or not, but should be treated as though they are arbitrary. But the rules are not arbitrary,

they are the product of logical consequences. With deductivism we can create a system whose

axioms are arbitrary givens, the rules to derive truths are logical, and the resulting theorems are

derivable from these.

Similar to other formalist approaches, this sidesteps the metaphysical and epistemological

questions. Mathematical statements still have no meaning, they are not about anything. We

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come to know them by following logical consequences. For example we have the rule, known as

modus ponens, that if P is true then Q is true and we know P is true then consequently Q is also

true. P and Q don’t have to refer to anything, but the result is logically consistent. To answer

the applicability problem, one just needs to look towards interpretation. We interpret the axioms

to be true, and since the theorems are logical consequences, then of course the results will be

very applicable.

Hilbert’s program specifically is to formalize all of mathematics axiomatically and then

prove the consistency of these axioms. That is to produce a set of axioms from which all

mathematics can be derived. Then, Hilbert must show that these axioms do not lead to a

contradiction.

In order to pursue his program Hilbert defined the notion of the finitary. Hilbert held a

similar view to Kant. To illustrate his view consider the image:

1, 11, 111, 1111, 11111, 111111, …

There is structure to this pattern we see, in each iteration there is an extra line added. It is easy to

see how we can continue this pattern by adding one line to the previous collection indefinitely.

This structure is immediately know to us by our intuition but is not a mental construction

because it is of a physical object, the lines on the paper.

From this structure we can build the finitary numbers. Take the collection “111” and this

can be symbolized as “3”. And so “2+3” is the same as “11+111”. Since numbers are about this

structure we can make some meaningful propositions, for instance two plus three equals five. Or

using variables, there is a number n less than ten such that n is odd. These statements are known

to be true due to their relation to the structure. The last example used a bounded quantifier

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which is good because one can simply check all the possible solutions. But problems arise when

an unbounded quantifier is used, the possible solutions are infinite in size. Take for example the

case, for any a and b, a+b=b+a, is true. One way to know this to be true is to show it is

impossible to be negated. The negation for this statement, if the statement were false, would be

that there exists an a and b such that a+b=/=b+a. But one cannot check every a and b because

there are infinitely many. The truth of this statement with unbounded variables cannot be

verified by checking all possible solutions. In order to prove this statement, Hilbert uses ideal

mathematics of the infinitary.

Ideal mathematics is statements that are unverifiable using only finitary means. Ideal

mathematics would be everything not provable using only the finitary such as infinite sets and

the real number line. Ideal mathematics is useful for finitary arithmetic. The idea of infinity is a

useful concept, but for Hilbert, but not directly derived from the structure. The infinitary is

extrapolation using intuition from the structure. Then using infinitary methods, one could prove

statements that are useful for the finitary, such as the earlier example that for any a and b

a+b=b+a.

Using finitary arithmetic and ideal mathematics Hilbert argued that one can construct all

the axioms for the formal system of mathematics. Then the hope was that the finitary could

show the consistency of Hilbert’s program.

However, Hilbert’s formalism, as well as Frege’s logicism, are severely impeded by

Godel’s incompleteness theorems. The proofs of Godel’s theorems are generally accepted as

true and they pose a serious problem for formalising mathematics.

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The first theorem states, “in every formal system that satisfies assumptions 1 and 2 and is

consistent, undecidable propositions exist [...] and so too in every extension of such a system

made by adding a recursively definable consistent class of axioms,” (Godel 62). For the

purposes of my paper it's important to note that in any formal axiomatic system that is powerful

enough to capture arithmetic, there will always be statements that cannot be proven or disproven

from the axioms.

The second theorem goes even further stating, “if c be a recursive, consistent class of

formulae, then the propositional formula which states that c is consistent is not c-provable; in

particular, the consistency of P is unprovable in P,” (Godel 70). Again for the purposes of my

paper its important to note that any formal system that captures arithmetic cannot prove its own

consistency. We cannot prove that a formal system is consistent or disprove it, using only the

formal system (due to space, for a complete summary of Godel’s incompleteness theorems refer

to Raymond Smullyan’s book Godel’s Incompleteness Theorems).

The first incompleteness theorem, shows the impossibility of the first half of Hilbert’s

second question. There can not be a complete set of formal axioms to derive mathematics from

because there will always be undecidable statements. The second theorem, proves that no formal

system can show its own consistency, which is the second half of Hilbert’s second question. In

sum, with Godel’s incompleteness theorems, Hilbert’s program fails.

There are there other schools of philosophy of mathematics that attempt to avoid Godel’s

theorems. First, is a return to Kantian ideas, a view called mathematical intuitionism. Next, so

called Structuralists are inspired by an earlier idea of Hilbert. Finally, Haskell Curry argues for a

revised version of term formalism.

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Luitzen Brouwer developed mathematical intuitionism, and was very much inspired by

Immanuel Kant. In his essay Intuitionism and Formalism he writes:

In Kant we find an old form of intuitionism, now almost completely abandoned,

in which time and space are taken to be forms of conception inherent in human

reason. For Kant the axioms of arithmetic and geometry were synthetic a priori

judgments, i. e., judgments independent of experience and not capable of

analytical demonstration; and this explained their apodictic exactness in the world

of experience as well as in abstracto. (Brouwer 83)

As opposed to Frege’s idea that mathematical truths are analytic, Brouwer agreed with Kant’s

argument that mathematical truths are synthetic a priori truths. That is they are relations of ideas

and cannot be know through definitions, so they are synthetic. They are a priori because they are

not dependent on sense experience. The problem rises that Kant situated mathematics in our

perception of space, mathematics is in some sense a science of measurements. We have an

innate understanding of space and so we can have an understanding of measurements. Then we

can know that one unit plus another equals two units. But with the development of non

euclidean geometry, this perception was shown to be false. For example in euclidean geometry,

the interior angle sum of any triangle is 180 degrees but in hyperbolic space it is strictly less than

180 and in elliptic it is strictly greater than 180. Kant argued that space is a projection of our

minds and we necessarily construct it as euclidean space. But there is evidence that space is not

necessarily euclidean but hyperbolic. In response Brouwer decides to situate mathematics in our

perception of time rather than space. Mathematical truths are mental constructions from our

understanding of time.

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Although space prevents me from giving this view proper attention, there are two reasons

I do not accept it. An argument I agree with is explained by Oliver Johnson in his essay Denial

of the Synthetic ‘A Priori.’ With the development of general relativity, special relativity, and

quantum mechanics, many so called synthetic a priori claims have been effectively challenged.

This shows that the world is not merely a construction of the mind. General relativity shows that

the mental construct of time is different from the nature of time. A modified version of the

applicability problem applies here as well. If mathematics is merely a construction of our mind,

then why is mathematics so applicable to the science?

Another view to avoid Godel’s incompleteness theorems is structuralism. Structuralism

is the view that mathematics is about patterns and structures. The number three is about its place

in the structure, it’s the third spot in the structure of natural numbers. Similar to Hilbert’s

finitary arithmetic, one comes to know a number by abstracting it from the structure. As Stewart

Shapiro explains the view, “most structuralist are realists in truth-value, holding that each

unambiguous sentence of, arithmetic and analysis, is true or false, independent of language,

mind, and social conventions of the mathematician,” (Shapiro 257). Structuralism can avoid the

earlier Julius Caesar problem because numbers are positions in the structure and Julius Caesar is

not a position in the structure. The number is not an independent object but contingent on its

place in the structure.

Though due to the limit space of the paper, I’ll briefly state my problems with

structuralism. The heart of the problem is that structuralism still seems to rely on abstracta, just

as logicism did. When asked what a number is, the response is its place in the structure. The

structure is the most reduced form. So when asked in the structure of natural numbers what is

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the number three, the response is in between two and four. Now it seems that the structuralists

still have yet to define what a number is. Then they may point to a physical structure of the

world such as Hilbert's strokes, the number three is one stroke more than two strokes. But the

number three is not just a direct referent to the strokes, the number three refers to “111” but also

“ooo”. Structuralism is inadequate for describing numbers..

Another way around Godel’s incompleteness theorems, the view which I will be focusing

on, is a revised version of term formalism by Haskell Curry. Weir describes the view in the

Stanford Encyclopedia of Philosophy that, “no restrictions are placed on what form the axioms,

rules and therefore theorems of a formal system are to be. Truth for elementary propositions of a

formal system consists simply in their provability in the system,” (Weir). Formal systems can

take any form, far reaching to very small in scope, and what is true is just what is provable in the

given system. True statements do not describe the world but are more a description of the formal

system they were proven in. For example, using the axioms of Zermelo-Fraenkel set theory,

someone can prove that one is greater than zero. Because that statement was proven in

Zermelo-Fraenkel does not make it a true fact of the world but merely a description of something

internal to Zermelo-Fraenkel’s system. It is merely a reflection of the fact that

Zermelo-Fraenkel axioms are ones that can prove that one is greater than zero.

Shapiro points out that, “for Curry, [...] mathematics is an objective science and it has a

subject matter. He wrote that ‘the central concept in mathematics is that of the formal system’

and ‘mathematics is the science of formal systems,’” (Shapiro 169). Mathematics consists of

studying different systems. There are important criterion for accepting a formal system. First,

that there is some intuitive reasoning for accepting the system, though he does not make any

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metaphysical claims about that intuition. Second, that system must be consistent or

non-contradictory. Third, the theory must be useful. It important to note that consistency for

Curry did not mean same thing as it did for Hilbert. Curry does not require a system prove its

own consistency as he states, “I maintain that a proof of consistency is neither necessary nor a

sufficient condition for acceptability,”(Curry quoted by Shapiro 170). In this way Curry avoids

Godel’s second incompleteness theorem, that it is not possible for a formal system to prove its

own consistency. A system needs to be non-contradictory so if it arises at a contradiction it

should not be accepted. But that system does not need to prove its consistency.

For example consider naive set theory. Since it does not need a proof of consistency, we

could accept the system. But Bertrand Russell showed with his paradox that it results in a

contradiction. So there exists a contradiction within the system of naive set theory. In light of

this naive set theory should not be accepted.

As opposed to the earlier game formalism, that math has no semantics, Curry’s formalism

is about something, it is about the formal system. Rather than saying we know something about

a mathematical statement because of the formal system, we know about a formal system because

of the statement. Mathematics is not solely syntax with no meaning, it is about something, it is

about the formal systems. Curry’s term formalism is less concerned about truth and consistency,

and is more concerned with what is pragmatic.

Consider the earlier example of Zermelo-Fraenkel set theory. The axioms may or may

not be true. There exists statements that are independent of the axioms, such as the continuum

hypothesis. The continuum hypothesis cannot be proven or disproven with the given axioms.

There does not exist a proof of its own consistency. But this system is studied due to its

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interesting properties, for instance Zermelo-Fraenkel set theory avoids Russell’s Paradox.

Further, there exists the system of Zermelo-Fraenkel with the axiom of choice. A proof of

something in Zermelo-Fraenkel with the axiom or choice is not a critique of Zermelo-Fraenkel

without the axiom of choice, but a description of Zermelo-Fraenkel with the axiom of choice.

For example, with the axiom of choice one can prove the well ordering of the real numbers. If

you are interested in calculus then you should use Zermelo-Fraenkel with the axiom of choice

rather than without the axiom of choice.

I argue that though Curry was able to revise formalism as to avoid Godel’s

incompleteness theorems, Curry still fails to address the earlier problem brought up by Frege

against term formalism: Why are these accepted formal system so applicable for the sciences?

Consider the example of the game of chess. Chess is a formal system, you have your

given pieces and your rules to follow, and you manipulate the game from there. But scientists do

not use chess in the sciences in the same way that they use mathematics. Chess is not about

more than the system itself but mathematics seems to be about more than just its system.

This application is best explained by mathematics having a meaning external to the

formal system. If mathematics is describing a feature of the world then it should apply to other

fields that describe the world. If mathematics is not meaningful, in the manner Curry proposes,

then mathematics should not be so applicable

An immediate objection is from Curry’s pragmaticism. The formal systems we study are

accepted because they are useful. We are interested in science and we form our formalized

system’s axioms and syntax based on that. The reason mathematics is so applicable to the

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sciences is due to our own interest in formal systems that would be applicable. The formal

system is constructed in order to help us understand science, and there is nothing external to that.

But that objection does not respond to mathematics applicability to other parts of

mathematics. Curry claims that there are multiple formal systems that describe different

branches of mathematics. There is a formal system for set theory, arithmetic, analysis, and

geometry. Truths proven in one of these fields is, however, useful to other fields. Why should

our system for analysis help with our system of geometry? The best explanation for this

applicability across systems is that they both have the same meaning external to the system.

This debate is similar to the debate between scientific realists and instrumentalists.

Realists argues that science determines metaphysical properties of the world. Instrumentalists

argue, on the other hand, that the scientific method is merely a useful tool. A common example

to illustrate this is the case of the electron. Electrons are unobservable but we have a theory of

electrons. The theory of electrons assists in our understanding of other theories and allows us to

make more accurate scientific predictions. In order to explain the ability of the theory of

electrons to make such good predictions the realist would argue that the theory is accurately

describing a feature of the world, and so electrons for the realist would actually exist.

Instrumentalists on the other had would argue that the theory is useful for our predictions but we

can’t make any assumptions of its metaphysical properties, the usefulness of electrons does not

necessitate their existence. For the instrumentalists all there is is what’s observable. But is left

with the problem of having to explain the applicability of science, if it is about something that

does not exist. The realists’ answer to applicability is that these observable things do exist so the

theories of them should confirm experimental results.

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Frege and the logicists have a view that has the same advantages as the scientific realists.

Mathematical truths are logical truths which are analytic, in the Fregean notion that they tell us

something about the world. Studying mathematics will tell me something about the would. But

then arises the problem that if mathematics is about the world then it must be referring to objects

in the world. The problem becomes about how do we know these abstract objects. Curry’s view

is similar to instrumentalism, mathematics is merely a useful tool. Curry view has a good answer

to the epistemological questions but is still subject to the applicability problem.

The objects of mathematics for Curry are the formal systems. These are not features of

the world but our own construction. So there is not serious epistemological questions because

we construct the formal system. When we know about mathematical statements, we know about

the formal system we have made and not some immaterial objects.

If we accept logicism, and that mathematical statements are meaningful and about

something. We are left with the question: How do we come to know these abstract objects that

mathematical statements are about? Since mathematical truths are the same as logical truths

there is an explanation for the usefulness of mathematics in the sciences.

These abstract objects bring to mind a platonist idea. These abstract objects are things

that exist in some platonic heaven. It seems that we can’t construct a description of these objects

because they are immaterial, so they have no sensical properties. In that sense, these are very

strange objects. We do not have access to this heaven because our sense experiences only

provide access to the physical world. Plato himself struggled with this question and came to the

conclusion that we must innately know about these object and rediscover them through life.

Fitzpatrick 20

This view of the existence of abstracta is difficult to accept due to the mysterious nature

of these objects and our questionable ability to know about them. I argue for a neo platonist

view similar to that of Hale and Wright. We come to know about these abstracta through Hale

and Wright’s principle of abstraction in relation to Hume’s principle. Admittedly it’s not known

through a causal relationship like scientific objects. But, arguably, to know what a number is

one only needs to understand Hume Principle and abstract numbers from it. The understanding

of Hume’s principle is a priori, to understand the principle is to abstract and understand what a

number is. One understands what a number is when one understand that the truth of the

equinumerosity of Fs and Gs necessitates the truth of the number of Fs and the number of Gs is

equal. So our access to these abstract objects is through this a priori understanding of Hume’s

principle.

Though Curry’s epistemology and metaphysics still might be easier to accept, the

neoplatonist view of abstract objects helps to mitigate the problems with traditional platonism.

Curry’s failure to resolve the applicability problem is significant. There is no explanation for the

applicability of mathematics across sciences and fields of mathematics. Revising mathematical

theories doesn’t seem arbitrary due to interest but because it better captures the world. If

mathematics is a description of the world, then our revisions are due to better explaining the

world rather than due to interest. And so the applicability of mathematics is better explained by

logicism than by Curry’s formalism.

The applicability problem is more important to address than the epistemological

problems. As shown earlier, abstracta are more accessible and less mysterious than a tradition

platonist view. And so the epistemological problem is less significant. But an explanation for

Fitzpatrick 21

the applicability remains important. We are insearch of mathematical and scientific theories that

are applicable. It seems that that search will not be fruitful unless those theories are about

something. Our criterion for accepting a theory is more than just its applicability, the acceptance

is due more to its explanation. The acceptance is due to answering the question: Why is it so

useful? The best explanation for the usefulness is that it is more accurately capturing the world

than other theories. The applicability problem is more urgent to answer in the debate between

logicism and formalism.

In sum, logicism has a better solution to the applicability problem than Curry’s term

formalism. Though at first the existence of abstracta raises epistemological questions, the

abstraction principle demystifies the nature of these mathematical objects. And so, one should

opt for a neologicist view rather than Curry’s term formalism.

Fitzpatrick 22

Works Cited

Brouwer, Luitzen. “Intuitionism and Formalism.” Inaugural Address at the University of

Amsterdam , 1912.

Frege, Gottlob. The Foundations of Arithmetic: a Logico-Mathematical Investigation into the

Concept of Number. 1884, Wesleyan University,

sshieh.web.wesleyan.edu/wescourses/2013f/388/e-texts/Frege%20Selections%20from%2

0The%20Foundations%20of%20Arithmetic.pdf.

Frege, G. (2013). Basic Laws of Arithmetic. Oxford University Press, Oxford. Translated by

Philip A. Ebert and Marcus Rossberg.

Godel, Kurt. On Formally Undecidable Propositions of Principia Mathematica and Related

Systems. Dover Publications, 1962.

Hale, Bob, and Crispin Wright. “Benacerraf's Dilemma Revisited.” European Journal of

Philosophy, 2002

Heck, Richard G. “The Development of Arithmetic in Frege's Grundgesetze Der Arithmetik.”

The Journal of Symbolic Logic, vol. 58, no. 2, 1993, pp. 579–601. JSTOR,

www.jstor.org/stable/2275220.

Johnson, Oliver A. “Denial of the Synthetic ‘A Priori.’” Philosophy, vol. 35, no. 134, 1960, pp.

255–264. JSTOR, www.jstor.org/stable/3748916.

Kant, Immanuel, and J. M. D. Meiklejohn. The Critique of Pure Reason. Generic NL Freebook

Publisher. EBSCOhost,

search.ebscohost.com/login.aspx?direct=true&db=nlebk&AN=1085932&site=eds-live&s

cope=site. Accessed 28 Apr. 2019.

Fitzpatrick 23

Shapiro, Stewart. Thinking about Mathematics: The Philosophy of Mathematics. Oxford

University Press, 2000.

Smullyan, Raymond M.. Gödel's Incompleteness Theorems, Oxford University Press,

Incorporated, 1992. ProQuest Ebook Central,

https://ebookcentral-proquest-com.library.emmanuel.edu:8443/lib/emmanuel/detail.actio

n?docID=271557.

Weir, Alan, "Formalism in the Philosophy of Mathematics", The Stanford Encyclopedia of

Philosophy (Spring 2015 Edition), Edward N. Zalta (ed.), URL =

<https://plato.stanford.edu/archives/spr2015/entries/formalism-mathematics/>.

Zalta, Edward N., "Frege’s Theorem and Foundations for Arithmetic", The Stanford

Encyclopedia of Philosophy (Winter 2018 Edition), Edward N. Zalta (ed.), URL =

<https://plato.stanford.edu/archives/win2018/entries/frege-theorem/>.