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AxiomathesWhere Science Meets Philosophy ISSN 1122-1151 AxiomathesDOI 10.1007/s10516-013-9211-x

On Some Properties of Humanly Knownand Humanly Knowable Mathematics

Jason L. Megill, Tim Melvin & Alex Beal

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ORI GIN AL PA PER

On Some Properties of Humanly Known and HumanlyKnowable Mathematics

Jason L. Megill • Tim Melvin • Alex Beal

Received: 15 October 2012 / Accepted: 1 March 2013

� Springer Science+Business Media Dordrecht 2013

Abstract We argue that the set of humanly known mathematical truths (at any

given moment in human history) is finite and so recursive. But if so, then given

various fundamental results in mathematical logic and the theory of computation

(such as Craig’s in J Symb Log 18(1): 30–32(1953) theorem), the set of humanly

known mathematical truths is axiomatizable. Furthermore, given Godel’s (Monash

Math Phys 38: 173–198, 1931) First Incompleteness Theorem, then (at any given

moment in human history) humanly known mathematics must be either inconsistent

or incomplete. Moreover, since humanly known mathematics is axiomatizable, it

can be the output of a Turing machine. We then argue that any given mathematical

claim that we could possibly know could be the output of a Turing machine, at least

in principle. So the Lucas-Penrose (Lucas in Philosophy 36:112–127, 1961; Pen-

rose, in The Emperor’s new mind. Oxford University Press, Oxford (1994)) argu-

ment cannot be sound.

Keywords Mathematics � Turing machines � Computability theory �Lucas-Penrose argument

J. L. Megill (&)

Department of Philosophy, Carroll College, Helena, MT, USA

e-mail: fourthantinomy@gmail.com

J. L. Megill

1803 Poplar Street, Helena, MT 59601, USA

T. Melvin

Department of Mathematics, Carroll College, Helena, MT, USA

A. Beal

Gnip, Boulder, CO, USA

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DOI 10.1007/s10516-013-9211-x

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1 Humanly Known Mathematics

Consider the set of humanly known mathematical truths at some particular random

minute m throughout human history; e.g., this minute, or the minute ten minutes

ago, or even the minute ten minutes from now. Again, this is the set of mathematical

truths that humans know at any given moment. Call this set ‘‘KMT,’’ for ‘‘known

mathematical truths.’’ The cardinality of this set will change—and specifically

grow—as we come to know novel mathematical truths. But given some fairly

reasonable assumptions, the cardinality of this set will be finite for all random

minutes m, if m is in the past, the present, or even the distant future. Numerous

arguments can be given for this claim. For instance, consider some random person

P alive at some random minute m. P will live a finite amount of time. So, P will only

have at most a finite number of thoughts. Furthermore, it seems that in order for us

to know a mathematical claim, we at least have to be able to think the claim.

Therefore, the set of mathematical truths that P will ever know is finite. And the

same will hold for all people. Say that the set KMT at a minute m is the set of

mathematical truths that everyone alive at or before m knows or knew at or before

m.1 KMT will be the union of a finite number (because the number of people in

existence at or before m will be finite) of finite sets (because every person will live

at most a finite amount of time and so will know only a finite amount of

mathematics), and so will be finite as well.2 In sum, the set of humanly known

mathematical truths at some random minute m will be finite.3

Moreover, if the cardinality of KMT is finite for any random minute m, then it

will be finite for all minutes m. This follows with general conditional proof, a basic

inference in predicate logic: if something holds of any random element in a domain,

1 We include mathematical truths that were once known but are forgotten at m (if there are any) in the set

of humanly known mathematical truths at m. This issue has no bearing on the argument.2 Note that once one picks a random point on the number line (a natural number), there will be only a

finite number of natural numbers between 0 and this number. It is the same with picking a random minute

m in human history; once one picks a minute, there will be a finite number of minutes between the

beginning of human history and this minute. So there will only be a finite number of people thinking a

finite number of thoughts up to that point in human history, even if human history extends infinitely into

the future.3 Here is a slightly different argument for the claim that KMT has a finite cardinality.

(1) Any particular random positive integer on the number line is a finite number.

For example, 737, 1098, 11123789…these are all particular integers and none of them are infinitely

large. Of course, this will be true of any positive integer. But then,

(2) Likewise, at any particular random minute m throughout human history, humans will only have

existed for a finite number of minutes and so for a finite amount of time.

One can simply assign each minute in human history a positive integer: we can call the first minute of

human history ‘‘m1,’’ the second minute of human history ‘‘m2,’’ and so on. Any particular random

minute throughout human history will be associated with a finite positive integer, and so up to that

minute, humans will only have existed for a finite amount of time. Therefore,

(3) At any particular random minute m throughout human history, the total number of humanly known

mathematical truths is finite.

Thoughts (and so mathematical thoughts in particular) take time to occur, so if there is only a finite

amount of time for them to occur, the number of them must be finite. But if we have to think a claim in

order to know it, then the number of human known mathematical truths at any given moment will be

finite.

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then it holds of all elements in the domain. Therefore, the set of humanly known

mathematical truths will be finite (even if very large) at any random minute

throughout human history. So, consider this finite set of all humanly known

mathematical truths at any minute m. This set, since it is finite, will be recursive; all

finite sets are recursive. That is, at least in principle, there will be an algorithm or

program which, when given a random string of characters, will give the value of ‘‘000

if the character string is in KMT and the value of ‘‘100 if the character string is not in

KMT; see, e.g., Wang (1981: 242), ‘‘A set is recursive if there is a program which

prints out 0 for inputs belonging to the set and 1 for inputs not belonging to the set.’’

Indeed, we can produce an algorithm. Consider a random string of mathematical

characters, which we will call ‘‘t,’’ the target string, and the set of humanly known

mathematical truths at any minute m. This algorithm searches for t in KMT. (1) Start

with n = 1, where n keeps track of the current element in KMT. When n = 1, we

are looking at the first humanly known mathematical truth. When n = 2, we are

looking at the second, and so on. (2) Look at the nth element. (3) If the nth element

doesn’t exist, return 1 (i.e., you’ve reached the end of the list). (4) If the nth element

is equal to the target string, then quit and return 0 (the target string is in the list). (5)

Failing the above two conditions, go back to step 2 with n = n ? 1 (i.e., look at the

next element). We are basically scanning through the finitely long list one element

at a time until we find the input on the list or reach the end of the list.4 In sum, the

set of humanly known mathematical truths at any random minute m is recursive. But

also note that for all sets, if a set is recursive, then the set is recursively enumerable.

This is an uncontroversial result in computability theory. See, for example, Wang

(1981: 242), ‘‘Every recursive set is recursively enumerable.’’ So, given that the set

of humanly known mathematical truths at any random minute m is recursive, it is

recursively enumerable.

The algorithm given above, when combined with various uncontroversial results

in the theory of computation, generates various results. Again,

(1) The set KMT at any minute m is recursively enumerable.

4 This algorithm is basically what computer scientists call a ‘‘linear search.’’ One can give a recursive

definition for linear searches. Call f the search function which takes in two parameters. (i) t, the target

string and (ii) l, the list we’re searching. f returns 1 if t is in l, otherwise it returns 0. This function also

uses the head() and tail() functions. The head function takes a list and returns its first element. The tail

function takes a list and returns everything but its head. So:

head 1; 2; 3½ �ð Þ ¼ 1

tail 1; 2; 3½ �ð Þ ¼ 2; 3½ �Also,

Nil is the empty list; so Nil ¼ ½ �As for f, there are three cases.

(1) l is the empty list (Nil). t cannot be an element of the empty list, so we return 0.

(2) We check the head of the list. If the head equals the target string, then we return 1.

(3) The recursive case, where we define f in terms of itself. Here we apply f to the target string (t) and to

the *tail* of l. This is how we move down the list. If the first element doesn’t equal t, then we lop off the

head of l and apply f once more. We’ll either find t and return 1, or keep lopping off the head until we end

up with an empty list, and return 0. This process will be finite because the list is finite.

Also note that while there is no algorithm for enumerating arithmetical truths, that doesn’t mean there

cannot be one for enumerating humanly known arithmetical truths.

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Furthermore, Craig (1953) has proven that,

(2) For all sets, if the set is recursively enumerable, then the set is axiomatizable.

So, of course,

(3) If set KMT is recursively enumerable, then it is axiomatizable.

But then with (1) and (3) (modus ponens), we can infer that,

(4) KMT is axiomatizable.

Of course, Godel’s (1931) First Incompleteness theorem demonstrated that,

(5) Any consistent axiomatic theory capable of producing a moderate amount of

number theory will be incomplete.

It follows (from (4) and (5)) that,

(6) KMT is either inconsistent or incomplete.

Suppose that humans do know a moderate amount of number theory. Then, these

truths of number theory that humans know will be in set KMT. But all of the truths

of number theory cannot be (consistently) known by humans, because then it could

be shown that all of the truths of number theory are axiomatizable (see, e.g., result

(4) above), which contradicts Godel. So, it must be that there is at least one truth

concerning number theory that is not in KMT (and this will be true of any random

minute, and so of all minutes, in human history). Therefore, human mathematics is

and will remain incomplete if consistent. One can generalize Godel’s theorem to

apply not simply to any particular formal mathematical system that contains enoughnumber theory, but rather to known human mathematics at any given moment. To

elaborate, Godel showed that there will be at least one true arithmetical claim—the

‘‘G sentence’’—that will not be provable in any consistent axiomatization of

number theory. But humans can, of course, look and see the truth of the G sentence

for many systems (e.g., the system used in the Principia). (Of course, Godel himself

thought that humans had some sort of intuition into the mathematical realm that

allowed us to see the truth of at least some mathematical claims.) But claim (6)

generalizes Godel’s result beyond formal systems to all of humanly known

mathematics; at any given moment in human history, there will be at least one

G sentence that we do not know, assuming that human mathematics is consistent.

(This is a result that might very well have come as a surprise (and an unpleasant one,

at that) to Godel.)

To continue, recall that the set of mathematical truths that are humanly known is

recursively enumerable; that is, there is an algorithm (or an effective procedure or a

‘‘program’’) for listing all and only the members of the set. But this means that,

(7) KMT could be the output of a Turing machine, i.e., a computer.

In other words, Turing machines can produce the same mathematical truths as

output that humans have produced and will produce (at any random moment in

human history). In practice, at least, the mathematical abilities of a Turing machine

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and human mathematicians are equivalent and will remain so throughout human

history.

One might object: one of two things will happen, either human history will be

finite, in which case everything we’ve said thus far is true, or human history will

stretch infinitely far into the future. And if human history is infinite, then perhaps

KMT will be infinite at some point as well? But this possibility is beside the point,

in a sense. At any particular moment in human history, there will be a finite amount

of time between that moment and the start of human history (see above). And

trivially, any particular person that will exist will exist at one of these particular

moments. KMT will be finite at any particular moment…and we will never be able

to step outside of KMT and know a mathematical claim that is not in KMT, even if

human history is infinite. Just as any particular number on the number line is finite

even though the natural numbers are infinite, no one will ever know more than a

finite amount of mathematics even if human history stretches on forever. For any

human that ever exists, human mathematics will be finite, recursive, recursively

enumerable, and so axiomatizable and Turing computable, and also either

incomplete or inconsistent.

We conclude this section by discussing an additional possible implication, one

that concerns Hilbert’s program. Of course, ‘‘Hilbert’s program’’ was an influential

project (in the first three decades of the 20th Century) in the foundations of

mathematics initiated by the mathematician David Hilbert. Hilbert, whose general

position is called ‘‘formalism’’ in philosophy of mathematics, wanted to set

mathematics upon a secure foundation by axiomatizing all of mathematics and then

proving that this axiomatization is consistent.5 Godel’s Incompleteness theorems are

traditionally thought to have greatly undermined Hilbert’s program. For example,

the First Incompleteness theorem shows that any consistent axiomatization of

number theory will be incomplete; so either not all of mathematics can be captured

by a single axiomatic system (like Hilbert wanted) or else the system will be

inconsistent (which Hilbert obviously did not want)…either way, Hilbert cannot

have what he wanted. However, while one cannot have a consistent and complete

axiomatization of all of mathematics like Hilbert wished, we saw above that the set

of humanly known mathematical truths (at any given moment) can be axiomatized,

at least. One could call the search for the axioms of humanly known mathematics

‘‘epistemic formalism,’’ and claim that although Hilbert’s program cannot succeed

in the way he wished, perhaps epistemic formalism could? That is, (a) humanly

known mathematics and (b) mathematics itself are two different things, and some

claims might hold of (a) (e.g., it is recursively enumerable) that do not hold of

(b) (because it is transfinite, it is not recursively enumerable). So even though, say,

Hilbert’s program cannot succeed for (b), it might be possible for it to succeed for

(a).6

5 See, for instance, (Zach 2009: introduction), ‘‘[Hilbert’s program] calls for a formalization of all of

mathematics in axiomatic form, together with a proof that this axiomatization of mathematics is

consistent.’’ Hilbert formulated his program across a series of works (e.g., Hilbert (1918)). See Peckhaus

(1990) for the historical development of Hilbert’s views.6 One might wonder if there could be a given mathematical truth T that is such that there is a ‘‘proof’’ of

T, but this proof is not recursive and undecidable. If so, then since KMT consists of recursive functions,

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2 Humanly Knowable Mathematics

Finally, consider the set of mathematical truths that humans can know. This set is

not equivalent to KMT. This set will contain all mathematical truths that any and all

people will ever know, for if someone knows a mathematical truth, it must be

possible to know that truth (see, e.g., modal logic system T, one theorem of which is,

‘‘if p, then possibly p’’). So it will contain KMT as a subset. But it will also contain

mathematical truths that we could know, but for whatever reason, never will know

(e.g., perhaps there is some large natural number that no one bothers to think about,

but if someone did, they could ascertain that this number is prime etc.). Call this set

‘‘PKMT’’ (for ‘‘possibly known mathematical truths’’).

But note that any given mathematical truth in PKMT could be or could have beenin KMT, even if it just so happens that it isn’t. This is true by definition: PKMT is

the set of humanly knowable mathematical truths. That is, PKMT is the set of

mathematical truths that humans could possibly know. So by hypothesis, any givenclaim in PKMT at least could be in set KMT. If a mathematical truth is possibly

known, then it is possible that it is known, i.e., it is possibly in KMT. So, e.g., at

least in principle, any random mathematical claim that is not currently in KMT but

is in PKMT, could be added to KMT. And the same results discussed above will still

obtain: KMT would still be recursively enumerable, axiomatizable, incomplete if

consistent, and potentially the output of a Turing machine. But then it follows that

any given mathematical claim that we could know could be the output of some

Turing machine or other. In other words, Turing machines can produce the same

mathematical truths as output that we could. Some arguments try to falsify

mechanism in mind by pointing to mathematics that we know or could know but no

Turing machine could; of course, the most prominent argument that does this is the

Lucas-Penrose argument.7 But it’s clear that if computers can produce the same

mathematical output that we (not only will but) can, then,

(8) The Lucas-Penrose argument cannot be sound.

(8) also entails that human mathematicians have certain limitations, just as Turing

machines do. For example, any Turing machine that can compute a minimal amount of

number theory will have a G sentence. If our possible mathematical output is Turing

computable, at least in principle, and Turing machines have a G sentence, then possible

human mathematics will have a G sentence. Indeed, this possibility has been raised as an

Footnote 6 continued

T would not be in KMT. But if we have this proof, then we would know T, so it seems T would be in KMT.

So T would be in KMT and not in KMT, an absurdity. It would be question begging here to simply assert

that there cannot be proofs that are not recursive (in a sense this is part of what we are trying to show). But

perhaps if our arguments that KMT is finite are correct, then a non-recursive proof of a claim would be

impossible? Just as a given mathematical claim can be thought of as a string of symbols, a proof can be

thought of as a string or collection of symbols (in which the end of the string is the claim being proven

etc.). Since our mathematical output must be finite (see the arguments above), it seems that this proof of

T, if we can indeed have such a proof, must be finite as well. But then the proof must be recursive as well.

So we cannot have access to non-recursive proofs, and the reason is our finite natures?7 For more on the Lucas-Penrose argument, see Lucas (1961; 1990; 1996), Penrose (1989; 1994) and

citations removed for blind review.

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objection to Lucas-Penrose in the past.8 The argument above is, in part, an attempt to

provide a rigorous demonstration that this possibility is in fact an actuality.

Of course, the Lucas-Penrose argument is not the only extant anti-mechanism

argument. Even assuming that our argument shows that Lucas-Penrose cannot be

sound, one might reasonably wonder if our argument can successfully reply to these

other arguments. For example, McCall (1999) offers an intriguing anti-mechanism

argument; roughly, the argument is:

A Turing machine can only know what it can prove, and to a Turing machine,

provability would be tantamount to truth. But Godel’s theorem seems to imply

that truth is not always provability. The human mind can handle cases in

which truth and provability diverge. A Turing machine, however, cannot. But

then we cannot be Turing machines. (Megill 2012: section 5)

One possible response is: Godel’s theorem shows that some formal systems—

such as the system used in the Principia Mathematica—are incomplete if consistent.

In any (consistent) system that contains enough number theory, there will be at least

one true yet unprovable statement; so truth and provability cannot be equivalent in

that system. But it might be that truth and provability are not equivalent for us as

well. To explain, perhaps we can see the truth of the G sentence for a particular

formal system S, even though the sentence cannot be proved in S. But perhaps our

(mathematical) cognition is an instantiation of a different formal system C. G cannot

be proven in S, but it can be proven in C; this is why we are able to see that G is true.

Moreover, as argued above, C will have its own G sentence; we will not be able to

prove or see the truth of this sentence. So, truth and provability might splinter apart

for us as well; we are no different from Turing machines in this respect?9

References

Benacerraf P (1967) God, the Devil, and Godel. Monist 51:9–32

Boyer D (1983) J. R. Lucas, Kurt Godel, and Fred Astaire. Philos Q 33:147–159

8 Benacerraf (1967), for example, suggests that perhaps humans have their own G sentence just like

Turing machines do? So when Lucas claims that we are different than machines—and so mechanism is

false—because we can see the truth of the G sentence, Benacerraf points out that perhaps we too have a

G sentence. See also Hofstadter (1979/1999).

Furthermore, there was an exchange between Boyer (1983) and Lucas (1990) that is relevant to our

claims. Boyer (1983) notes, as we do in connection with the set KMT of humanly known mathematical

truths at some random minute m, that any human mind will have a finite amount of output and so this

output could be produced by a Turing machine. Lucas (1990) thinks that this objection misses the point,

because what he is concerned with is the idealized case, i.e., with what humans and machines can do in

principle, once we ignore limitations on us that arise because of our finitude etc. But if we are correct that

any given member of the set of humanly knowable mathematical truths could be the output of a Turing

machine, than we have an effective counter to Lucas’s response to Boyer.9 Another anti-mechanism argument is formulated in Cogburn and Megill (2010). They argue that ‘‘if

Intuitionism (in philosophy of mathematics) is true, then we are not Turing machines.’’ Perhaps we could

simply argue, however, that since our argument shows that we are Turing machines, Intuitionism (or at

least one of the core tenets of it that Cogburn and Megill use to formulate their argument) must be false?

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Cogburn J, Megill J (2010) Are turing machines platonists? Inferentialism and the computational theory

of mind. Mind Mach 20(3):423–440

Craig W (1953) On axiomatizability within a system. J Symb Log 18(1):30–32

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Hilbert D (1918) Axiomatisches Denken. Math Ann 78:405–415

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Brighton on April 6th. http://users.ox.ac.uk/*jrlucas/Godel/brighton.html

Lucas JR (1996) The Godelian argument: turn over the page. A paper read at a BSPS conference in

Oxford. http://users.ox.ac.uk/*jrlucas/Godel/turn.html

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Penrose R (1989) The emperor’s new mind. Oxford University Press, Oxford

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University Press. Second edition, 1925 (Vol. 1), 1927 (Vols 2, 3). Abridged as Principia

Mathematica to *56, Cambridge: Cambridge University Press, 1962

Zach R (2009) Hilbert’s Program. The stanford encyclopedia of philosophy (Spring 2009 Edition),

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