The pragmatist approach, as stated in this essay,
takes into account two features of knowledge, both having an enormous
potential of growth: the scope of science, whose frontiers can be
infinitely advanced, while firmness of its propositions grows
with consolidating once attained frontiers. An opposite view may be
called limitativist as it conservatively sticks to some a priori
limiting principles which do not allow progressing in certain
directions. Some of them influence science from outside, like ideological
constraints, other ones are found inside science itself.
The latter can be exemplified by the principles like
those: (1) there can be no action at a distance; (2) there are no necessary
truths; (3) there are no abstract objects. The first might have happened to
limit physics with rejecting the theory of gravitation. The second entails
that arithmetical propositions are either devoid of (clasical) truth or are
not necessary; this would limit arithmetic to the role of a mere calculating
machine, without giving any insights into reality. The third principle, for
instance, limits logic to the first-order level (since in the second one
variables range over abstract sets). The history of ideas shows that such
limiting principles, had they been obeyed, would have hindered some great
achievements of science. This is why we should not acknowledge any such
principle as necessarily true, that is, winning in confrontation with any
view contrary to it. Such principles on equal terms should compete with
other propositions in obtaining as high degree of epistemic necessity as
they may prove worth of.
To the core of pragmatist approach there belongs treating
epistemic necessity as a gradable attribute of propositions. In
accordance with ordinary usage, "necessary" is a gradable adjective, having
a comparative form. The degree of epistemic necessity of a scientific
statement depends on how much it is needed for the rest of the field of
knowledge (Quine's metaphor). The greater damage for knowledge would be
caused by getting rid of the point in question, the greater is its epistemic
necessity. At the top of such a hierarchy are laws of logic and arithmetic.
Among physical laws at a very high level there is the law of gravitation,
owing both to its universality, that is, a colossal scope of possible
applications (advancement of frontiers), and its having been empirically
confirmed with innumerable cases (consolidation of frontiers). Such a
success has proved possible owing to the bold transgression of the limiting
principle 1 (see above), and this has resulted in so high a degree of
The pragmatist approach of this essay can be
expressed with the Chinese proverb: Black cat or
white cat: if it can catch mice, it's a good cat.
Another motto is suggested in the announcement of a
scientific conference: The frontiers of science are
by definition continually shifting. Such a continuous shifting is
what we call advance of science.
§1. Frontiers versus limits
The kinship of meaning between these terms is misleading,
though in some translations both happen to be rendered with the same word,
e.g. German "Grenze", Polish "granica". However, there is a significant
opposition in their use.
A limit is something static and negative -- to mark the
line that is not allowed, or not likely, to be gone beyond. When, for
instance, we speak of a sequence of numbers as tending to a limit, we mean a
point that cannot be exceeded; the sequene definitely stops at this point
(while we do not speak of a number sequence as tending to a frontier). The
derivative "limitation" means setting a limit to changes, in particular,
changes being progressive.
On the other hand, "frontier" means either a dynamic line which can shift
forward, or a region to be occupied owing to such a shift.
Definitions of frontier as found in dictionaries
are as follows.
- A region just beyond or at the edge of a settled area.
- A wilderness at the edge of a settled area of a country.
- An undeveloped field of study; a topic inviting research and
Meanings 1 and 2 have evolved in the idiom of American
settlers and pioneers at the Wild West, who had moved forward with their
horses and wagons. Item 3 refers to an intellectual quest of pioneers and
discoverers in science. Such an advancing frontier marks successive
In human actions, the existence of a limit
means a constraint to stop some moves or prevent some kinds of behaviour.
This is something that makes narrower the scope of our freedom or our
possibilities. There may be limitations imposed on human actions through
some human decisions; this is the case with legal systems, monastic rules,
military discipline, etc.
Moreover, there are limitations which derive from the
natural order, and get perceived and recognized by people. These are usually
expressed in the form of rules to control our behaviour. If such a rule is
of special importance, somehow fundamental, it is often honored with the
name of a principle. Thus we come to the point that there is an
important category to deserve the name of limiting principles.
The choice of this name is no excentric novelty. Already
in 1949 it was introduced to the philosophical vocabulary by the famous
British philosopher C.D.Broad. Here is his definition.
"There are certain limiting
principles which we unhesitatingly take for granted as the framework
within which all our practical activities and our scientific theories
are confined. Some of these seem to be self-evident. Others are so
overwhelmingly supported by all the empirical facts which fall within
the range of ordinary experience and the scientific elaborations of it
[...] that it hardly enters our heads to question them. Let us call
these Basic Limiting Principles."
See: "The Relevance of Psychical Research to Philosophy",
Philosophy 24, pp. 291-309.
I take here advantage of invoking a well-known author, but
I do not follow his own list of limiting principles. Broad was most
interested in mind-body relations, hence his principles mainly deal with
that domain. Here we need a more comprehensive use to involve various
domains of science and philosophy. The ordinary meaning of the verb "to
limit" makes such a broad use justifiable. Hence I employ the phrase
"limiting principles" to denote constraints exercised on our knowledge from
outside, by some institutions or ideologies (example  in §2), as well as
those acting within philosophy or science.
Nevertheless, the claim LP.2 was essential in the
original research of Broad; in the period about 1920, together with
Betrand Russell, he belonged to that small circle of philosophers who
understood revolutionary ideas of then current physics.
www.hist-analytic.org/russell_and_broad_on_space_apa.htm, Russell and C.
D. Broad on Space by Steve Bayne, 2000, Bertrand Russell Society,
The fact of being subjectively taken for granted does not
necessarily render such principles objectively true. Some of them might be
right, other ones wrong. If a limiting priciple is right, then it helps us
to avoid errors, otherwise it puts a limit to progress, that is, withholds
advancing frontiers of science.
§2. Some samples of
As instructive examples of such limitations concerning
science and philosophy, let us consider the following principles.
- LP.1: The LP that the teaching of Catholic Church
forms a source of limitative principles concerning development of sciece and
philosophy. This general limitative principle has been divided into quite a
number of detailed instructions in the basic document of 1864 entitled "The Syllabus of
Errors Condemned by Pius IX".
This document (www.papalencyclicals.net/Pius09/p9syll.htm)
lists opinions judged as erroneous, hence in order to learn a limitative
principle from any of them, the sentence in question should be denied. For
instance, when the condemned view (item 14) reads "Philosophy is to be
treated WITHOUT taking any account of supernatural revelation", the
replacement of the negative particle "without" by the positive "with" yields
the following (numbered as non-14) limitative principle:
- non-14) Philosophy is to be treated WITH taking into account
the supernatural revelation.
- non-11) The Church ["not" cancelled] ought to pass
judgments on philosophy, and ought NOT [added] to tolerate the errors
- non-12) The decrees of the Apostolic See and of the Roman
congregations DO NOT impede the true progress of science.
Let us imagine some limitations following from these
principles. As for 14, philosophy of mind could not be developed without
maintaining, for instance, the dogma of soul immortality; at this point the
freedom of research would be limited. According to 11, the freedom of
inquiries should get limited to those philosophical statements which are not
regarded by the Church as wrong. According to 12, it is not allowed to,
e.g., assert that the condemnations of Copernicus and Galileo impeded the
true progress (Copernicus' condemnation has been revoked in 1835).
- LP.2: The Leibnizian LP: There can be no action at
a distance. I call it Leibnizian (for mnemotechnic reasons) though
Leibniz was no alone to blame the idea of gravitation for violating the
principle in question. However, his eminence among the critics seems to
justify such naming.
- LP.3: The Humean LP, shared by the Vienna Circle:
No proposition concerning the reality outside language enjoys the status of
epistemic necessity, since any proposition is either empirical or
mathematical. Being empirical, it is refutable, hence not necessary. Being
mathematical, it has no epistemic import for it does not deal with any
reality; hence its necessity is a matter of linguistic convention unable to
grant any cognitive content to mathematical theorems.
- LP.4: The nominalist LP: higher order logics should
be disregarded for the lack of any objectual reference of their quantified
- LP.5: The constructivist LP: in order to acknowledge
the existence of a mathematical entity, it has to be constructed by
appropriate operations of human mind.
The first item represents limitations imposed on science
and philosophy from outside by authorities having had considerable means to
hamper intellectual quests. Why to mention such things nowadays, when in our
open society such restrictions have lost any compelling power? However,
there is an instructive moral in the story. Not so much in the publishing of
Syllabus in 1864, but in the fact that the present practice of Catholic
Church -- with respect to any research -- agrees with the claims having
been blamed in Syllabus. Since these claims derive from the philosophy of
Enlightenment, it may be said that nowadays we witness the Church converted
to Enlightenment (not at this point alone, also at the point of human
This deserves to be regarded as a success of pragmatic
attitude toward science. The Enlightenment belief in the power of reason was
mainly due to the astonishing success of Newton's physics, esp. his theory
of gravitation. This achievement consisted in a formerly unaimaginable range
of applications of a scientific theory. Applications which have extended
over the whole universe, from the earth to most remote stars, macroscopic
regions as well as microscopic ones. Such a pragmatist argument must have
convinced the whole academic world, and the whole educated public, about the
power of human reason even if acting against theological LPs. And then, the
only reasonable move to have remained to the Church was to retreat from
condemning the autonomy of science.
§3. Newton's gravitation as
a "good cat" to advance frontiers of science
The claim LP.2 was regarded by Broad as stating a
fundamental limiting principle. This remains in accord with what was
asserted by such eminent thinkers as, for instance, Leibniz. Nevertheless,
the overt transgression of that principle by Issac Newton with his theory of
gravitation is counted among the greatest achievements in the history of
science. An unimaginable set of phenomena grows explained by the simple
equation to state that the gravitional force is proportional to the product
of masses of the bodies in question, and inversely proportional to the
square of the distance between them. This is that force which plays a
decisive role in the whole cosmic scenario from the very beginning of the
Note, however, that it is a force which does exert an
instantaneous action at a distance, both features being
forbidden by the principle in question. This is why this idea was vehemently
objected by Leibniz. What more remarkable the same objections were troubling
Newton himself, nevertheless, it was his pragmatic attitude which took over
fundamentalist scrupples. In spite of his being deeply uncomfortable with
the notion of "action at a distance" which his equation implied, finally he
"It is enough that gravity does really exist
and acts according to the laws I have explained, and that it abundantly
serves to account for all the motions of celestial bodies"
Iitalics mine - WM - to stress the pragmatic attitude which has
won at last.
Thus the theory of gravitation has practically proved a
good cat, even if this cat might have appeared black, that is, undesirable
from a theoretical point of view. Had Newton yielded to Leibniz's attack and
his own reservations, then his enormously seminal theory, forwarding the
frontiers of science as much ahead as it never happened earlier, would have
fallen prey of a categorical limiting principle.
To follow a sequel of this story, one should go deeper
into Newton's doubts and Leibniz's charges. Let us take a look at the latter.
The very title of Leibniz's text reveals - in an ironic
vein - the main line of his argument. It reads: "Antibarbarus Physicus pro
Philosophia Reali contra renovationem qualitatum scholasticarum et
intelligentiarum chimaericarum". Here "barbarian" is to mean "uncultured
person", hence Leibniz sees himself as a defender of a higher intellectual
culture. This culture amounts to rejecting the scholastic way of thinking
characteristic of the Middle Ages (barbarian, in a sense).
Let me recall that schoolmen fancied occult qualities, or
occult forces, to explain phenomena, as in that satire by Molier in which a
scholastic doctor asked why opium makes one sleepy, explains quite
seriously: "for there is in it the force to make one sleepy". No knowledge
about reality ("philosophia realis") is conveyed by such ridiculously
superficial explanations. Ironically, Leibniz compares the force of
gravitation to such scholastic figments, and speaks against their revival,
that is, "contra revovationem qualitatum scholasticarum".
See: Die Philosophische
Schriften von Gottfried Wilhelm Leibniz Herausgegeben von
E.J.Gerhardt, VII Band, Georg Olms, Hildesheim 1961, pp.337-343, passim.
Instead, Leibniz calls for any concept being introduced
(here "gravitation") that it be defined in terms of some obvious primitives
notions of mechanics, namely those of magnitude, form and movement. These he
regarded as simplest and most obvious in the language of physics, and blamed
the idea of gravitation for its not being reducible to those conceptual
primitives. Newton had a similar research programme: in other cases he
successfully tried to explain the origin of various forces which acted on
bodies, but in the case of gravity, he did not succeed to identify any
motion producing the force of gravity.
If so, why Leibniz and Newton so much differred with each
other in their final conclusions? The deep difference lies in the
respectives philosophies of science. Newton's was spontaneously pragmatist
(though the term itself was not in use then), while Leibniz's was
fundamentalist, firmly sticking to limiting principles.
The point of this story? It evidences that in some crucial
questions it is pragmatism what moves the frontiers of science ahead,
sometimes up to a farthest attainable point, as it was the case with Newton.
The story has continuation in
Einstein's theory of general relativity, in which gravitation is an
attribute of curved spacetime instead of being due to a force propagated
between bodies (did this satisfy Leibniz's expectations?). This, however, is
a separete issue to be handled by historians of physics, esp. experts in
Another point in current physics related to action at a
distance, even more sophisticated, is that of Quantum
Entanglement. An extensive and lucid treatment of this subject,
including the problem of teleportation (which sounds like a story about
action at distance), together with Einstein's objections, are lucidly
explained in the article "Quantum Entanglement and Information" (2010) by
Arthur Fine, found in "Stanford Encyclopedia of Philosophy"
(plato.stanford.edu/entries/qt-entangle/). As quantum physics and quantum
information go to the furthest frontiers of current science, evidently these
themes are highly worth study.
There was a double enormous surprise in Newton's theory of
gravitation: the universality, extending over the whole universe, its
whole past and future, as well as the fact the new theory surpassed all
scientific achievements of antiquity; those up to the 17th century were
commonly regarded as insuperable. The latter feature has decidedly
contributed to that trust in human reason which were to mark the coming age
of Enlightenment. To conclude: note that this surprisingly efficient theory,
explaining the universe and forwarding the course of civilization, is much
due to Newton's pragmatic approach; thus pragmatism has proved its mettle
against an unconditional relying on limititative principles.
§4. Epistemic necessity of as a high degree of indispensability
This Section is to perform two interrelated tasks: (1)
first, to provide another case study of how a limiting principle may slow
down progress of science; second, tu use the same study for introducing a
concept which would deeper explain the process of advancing science, to wit
the concept of epistemic necessity as a gradable property of propositions.
The advancing of frontiers, say, in the policies of an
empire, consists of two actions: first, the conquering forces are to reach
toward a point in the terrain to be annexed; second, this new frontier
should get consolidated to secure it against the risk of being lost.
An intellectual conquest comprises two similar phases. In the case of the
law of gravitation it was (1) to propose this law as universal, ruling the
whole universe; (2) to gradually check its applications to various kinds of
phenomena, and various regions of the universe.
With each such application check successfully passed, this
law proved more and more indispensable for understanding reality. There
continually grows the number of phenomena which it explains and predicts.
Nowaday, for instance, we learn owing to it about the initial forming of
hydrogen from the plasma left behind from the big bang, about gravitational
callapses of stars, etc. "Those things in heaven" (to cite Hamlet) which
Newton could not have dreamt of, more and more extend the frontiers
of the known universe; at the same time, they increasingly confirm the
validity of the law, and this amounts to ever greater consolidation.
Both extension and consolidation combine into advancement of
The more proceeds such an advancing of the law in
question, the more it grows indispensable. Such a status of being
indispensable element of our knowledge deserves to be called epistemic
necessity. The adjective "indispensable" means something not to be
dispensed with, something that cannot be done away with.
When so defining "epistemic necessity" in terms of
"indispensability", one should make it clear whether or not the latter
admits a gradation. For it may happen that a product X which satisfies a
need perfectly, nevertheless can be replaced by a substitute Y. Should we
then deny indispensability to X? It depends on a comparative estimating of
their merits. Suppose that the substitute Y brings the same result but at a
greater cost: for instance, slower (expense of time), with an additional
risk, with less convenience, etc. Then we shall say that X is more
indispensable than its substitute Y. In this sense, indispensability
proves to be a property capable of being graded. And so gradable is
epistemic necessity of a proposition -- when defined in terms of its
indispensability for our knowledge.
When the concept of necessary proposition gets referred to
some objects, this challenges a limiting principle listed in §2, namely
LP.3. This principle claims the non-existence of necessary propositions
among those being concerned with any domain of reference. This limitation
derives from the empiricist contention that every proposition about the
world -- called synthetic for its adding a new piece to our
knowledge -- must be justified on the basis of sensory experience. Only then
it grows capable of being either true or false.
Otherwise, a proposition cannot pretend to be true. Such a
detachment from reality -- according to that view -- is characteristic of
mathematical propositions: their sole import for science consists in being
rules to transform strings of symbols into other strings in a process of
computing. If one calls them necessary, this is just in the sense of
necessity relative to a linguistic convention; 2+2 equals 2 in virtue of
certain conventions, termed meaning postulates, regarding the
meanings of symbols "+", "=", etc. In this approach, the necessity is
coextensive with the property of being analytic, and so there arises
the famous dychotomy synthetic-analytic. Analyticity is conceived as not
admitting of any gradation.
Had that Vienna Circle claim been taken seriously, this
would have blocked metamathematical research, for instance, inquiries into
completeness of the first-order logic or completeness of arithmetic. For
completeness means provability of all the truths in the theory in question,
hence it is assumed in such a research that mathematical propositions are
either true or false.
However, neither Kurt Gödel nor Alfred Tarski were much
impressed by this Vienna doctrine. Their studies have confirmed that
mathematical statements are capable of receiving the values of truth or
falsity. And so their epistemic necessity continues to be a point at issue.
This attribute is regarded by some philosophers as coextensive with being
a priori, that is, preceding, or being before (literal translation
of "a priori) any sensory experience.
A thorough analysis of the a
priori, frequently referred to in literature, is given with Morton White's
study "The analytic and the syntetic" in his book Toward Reunion in
Philosophy, Harvard University Press, Cambridge, 1956.
This view gives rise to the famous old controversy whether
mathematical axioms are necessary while not being analytic. The name coined
for such instances reads: "synthetic a priori". It is meant to express the
point that such sentences add a piece of information to our knowledge (so
being synthetic), but without being preceded by any sensory experience (so
being a priori). This debate appears far from conclusiveness, so intricate
are notions and assumtions involved.
Fortunately, the pragmatist approach is free from such
perplexities. Once taking for granted that epistemic necessity is gradable,
we encounter no question of either dychotomy or trichotomy. Instead, there
is a scale of epistemic necessity degrees. Let the totality of our knowledge
be represented by a field of force (as pictured by Quine). Points near edges
symbolize narrow generalizations; their removal would not disturb the rest
of field considerably, so readjustments would be relatively ease; this means
a law degree of indispensability. Being found in the interior, closer to the
center, means for a proposition to possess a broader field of applications
(extending up to the edges), hence to enjoy a greater indispensability.
Closest to the center are logical and mathematical statements; had they
disappeared, the whole structure would collapse, and require a total
reconstruction, a bulding anew (provided there be such a chance). These have
the rank of the greatest epistemic necessity.
Such a model of knowledge does not imply the existence of
an absolute necessity. Also in the circle closest to the center, some
revisions are not unthinkable. Even classical propositional logic happens to
be readjusted for some purposes, as seen in certain discussions about the
law of excluded middle. Anyway, propositional logic, as being decidable,
belongs to theories closest to the top of epistemic neceesity. Next to such
a top would be the predicate logic as having proofs of consistency and
completeness, but inferior to sentential logic for lacking decidability.
At that altitude there is the place for arithmetic, though
it does not possess the attribute of completeness. As for consistency, it
cannot be demonstrated with means which would exceed the inferential
capabilities of arithmetic itself; this can be done only with some means of
stronger systems, such as set theory, but those stronger ones, again, cannot
have proofs of consistency without using still stronger means (new axioms,
or new inference rules, which result in a greater ontological commitment,
e.g. acknowledging the existence of sets of sets). Nevertheless, we do
firmly believe in consistency of arithmetic on the strength of many
centuries of experience with applying it in innumerable cases. Had
arithmetic been inconsistent, then in such an enormously long time there
must have occurred an error in applications. To use again the Chinese
proverb we started with: if there is any cat which can catch mice with a
possible greatest efficiency, such an enormously good cat is arithmetic.
With such a pragmatist certificate, arithmetic propositions obtain the
status of the possibly highest epistemic indispensability.
Let me sum up this piece of discussion, even at the cost
of some repetitions, with quoting a text by W.V.O. Quine, which forms an
essential part of his pragmatist manifesto.
"Total science is like a field of force whose
boundary conditions are experience. A conflict with experience at the
periphery occasions readjustments in the interior of the field. Truth
values have to be redistributed over some of our statements.
Re-evaluation of some statements entails re-evaluation of others, because
of their logical interconnections -- the logical laws being in turn simply
certain further statements of the system, certain further elements of the
field. Having re-evaluated one statement we must re-evaluate some
See "Two dogmas of empiricism" in: From
a Logical Point of View, Harward University Press, Cambridge, Mass.
1953, p.42 (Section VI). See also: www.ditext.com/quine/quine.html.
Another Quine's metaphor tells us that the degrees of
necessity are like degrees of
greyness, instead of forming the
"The lore of our fathers is a fabric of sentences.
[...] It is a pale gray lore, black with fact and white with convention.
But I have found no substantial reasons for concluding that there are any
quite black threads in it, or any white ones."
This statement is found in his article of
1954: "Carnap and Logical Truth" contained in the volume The Ways of
Paradox and Other Essays, revised edition, Cambridge, MA: Harvard
University Press, 1976, pp. 107-32. This parable is discussed by Yemina
Ben-Menahem, "Black, White and Gray: Quine on Convention",
Synthese (2005) 146: 245-282.
§5. The inferential and computational power of higher-order logics
The limiting principles LP.4 and LP.5 (Section §2) deserve
special interest. Were they obeyed this would have a disastrous impact on
the progress of mathematics and computation. In considering the power of
higher-order logics, which are forbidden by LP.4, one should start from a
seminal statement by Kurt Gödel. In the paper "Über die Länge von Beweisen"
(on the length of proofs, 1936) he pioneered the following idea.  some
proofs, which in the first-order logic cannot be carried out (thus giving
rise to undecidability), can be carried in the second-order logic, and 
other ones which at the first-order level would require time not being
available either to humans or to computers, become tractable in an
accessible time when performed at the higher level. What, in turn, is not
tractable in the second-order system of logic, may prove tractable in a
third-order system, and so on.
In his short report Gödel did not give any proof of these
statements. The proof has been given much later by S.R.Buss.
Samuel R. Buss, "On Gödel's theorems on lengths
of proofs I: Number of lines and speedups for arithmetic." Journal of
Symbolic Logic 39 (1994), 737-756.
A fact much relevant for the issue in question is provided
with a remarkable exemplification of second-order logic's capability. It is
found in an article by Boolos.
See: George Boolos, "A curious inference",
Journal of Philosophical Logic 16: 1-12.
He gave a formalized proof of a certain arithmetic theorem
in the second-order logic. This took space of about one printed page, hence
several thousands single symbols.
On the other hand, in the first-order logic no formalized
proof gets tractable (i.e., computable in practice) either for Boolos or for
computer, since in any case it would require a number of symbol greater than
the number of atoms in the observable universe. Boolos estimated that this
quantity would be represented by an exponential stack in which a number is
raised to the second power 64536 times.
What about a formalized computer-assisted proof in the
second-order logic? In the print it has to be longer than Boolo's text
because of requirements imposed by the softare to check correctness. In the
literature at least two such proofs are presented having size of several
tens of printed pages what is, in fact, a tractable size. Both proofs, given
two different system of computer-aided reasoning, are found in the
Christoph E. Benzmüller and
Chad E. Brown, "The Curious Inference of Boolos in Mizar and OMEGA",
Studies in Logic, Grammar and Rhetoric, 10(23), 2007, special
volume From Insight to Proof. Festschrift in Honour of Andrzej
Trybulec edited by Roman Matuszewski, Anna Zalewska, University of
Bia³ystok, pp.299-386. On line:
The experience obtained by the said researchers in
performing the above task made it possible for them to estimate computer
capabilities with respect to a more difficult performance. Let us imagine
that a computer system is to be used not for checking a human-made
formalized proof, but for devising such a proof by itself. Let it be the
proof of the same theorem which was inquired by Boolos. The authors see the
problem as follows.
"Boolos' example perspicuously
demonstrates the limitations of current first-order and higher-order
theorem proving technology. With current technology it is not possible to
find his proof automatically, even worse, automation seems very far out
of reach. Let's first give a high-level description why this is so.
Firstly, Boolos' proof needs comprehension principles to be available
and it employs different complex instances of them. [...] Secondly,
the particular instances of the comprehension axioms cannot be
determined by higher-order unification but have to be guessed.
However, the required instantiations here are so complex that it is
unrealistic to assume that they can be guessed. [...] Here it is where
human intuition and creativity comes into play, and the question arises
how this kind of creativity can be realised and mirrored in a theorem
prover." [Italics - WM].
Christoph E. Benzmüller and Manfred
Kerber, "A Challenge for Mechanized Deduction", 2001. The Web page quoted
did not exist in the time of writing the present paper. The quotation is
rewritten from: Witold Marciszewski, "The Gödelian Speed-up and Other
Strategies to Address Decidability and Tractability", Studies in
Logic, Grammar and Rhetoric, 9(22), 2006, University of Bia³ystok,
pp. 9-29. On line: http://logika.uwb.edu.pl/studies/vol22.html.
The reference to the essential role of comprehension
principle makes us aware how much the second-order logic is here relevant.
Moreover the use of this logic requires intuition and invention unavailable
to computer systems; and are just the priviledge of human minds. Hence it is
up to humans to advance frontiers of knowledge far ahead. If only they be
bold enough to not observe limitative principles like that banning
Next, I am to pay attention to a curious fact about the
axiom of choice. In spite of various doubts and objections, this
statement proves essential and indispensable in automated theorem proving.
Hence its common practical acceptance in that circle of researchers.
This is connected with the procedure of skolemization,
that is, reduction to Skolem normal form. Owing to this procedure, a
reasoner gets rid of quantifiers, and thus the formula in question gets
transformed into an expression of sentential calcules. This, in turn, makes
it possible to apply an algoritmic decision procedure of this calculus. Thus
we are able to algorithmically establish whether the formula is, or is not,
a tautology of predicate logic. As is commonly known, such a procedure fails
in some cases. Sometimes, when the solution would be in the negative, the
algorithm falls into the loop, and never stops. Nevertheless, skolemization
(or something equivalent, e.g. Hilbert's episilon operation) is the most
efficient procedure for such partial decidability. It requires no guesses,
no invention or intuition, and thereby it can be performed by computers.
However, there is a philosophical cost of such a
convenience. We have to violate the limiting principle listed as item LP.5
in §3. This priciple is not respected by the axiom of choice. For no choice
function is defined in it to hint at the criteria of selecting
representatives of certain sets to form a new set out of them. The existence
of such a function is postulated without identifying its content. This is
supposition necessary for eliminating quatifiers in expressions of the form:
In such a simple case (just one universal quantifier)
skolemization is performed by replacing the existentially quantified
variable y with a term f(x). If there are more universal
quantifiers, then the function has correspondingly more arguments. In
performing such instantiation, we do not bother about defining or
constructing such a function, we simply assume that it does exist. Such
arbitrariness may be judged as reckless by philosophically cautious people
who prefer to observe the limiting principle LP.5. Nevertheless it renders
enormous services in research, and so advances the frontiers of our
§6. Pragmatic insights ("this should work") beyond common intuitions
The parenthesed phrase is to suggest what I mean under
pragmatic insight as compared with common intuition. This
comparison is needed in order to detect those sources of fallacies which
happen to be accepted as limiting principles. I consider here not only those
limiting principles which we find in scientific or philosophical literature,
but also those appearing in our everyday thinking.
The latter, even if not explicitly stated, limit our
understanding of the world. An instructive example is found in fairly common
intuitions concerning the free fall of bodies. In spite of passing exams in
school physics, there are educated people who believe that -- in any
conditions whatever -- heavier bodies are bound to fall faster than lighter
ones. Galileo and Newton were able to discard that erroneous perception
since they expected from the laws of nature an universal range of
applications; and this is hardly available for intuitions born from our
everyday experience. In the case in question our observations refer to
bodies falling down to earth in the earthly atmosphere which produces the
air resistance. In thus narrowed conditions, the impression of differences
about the speed of falling bodies is not misleading; however, without such a
restricting proviso there arises a fallacious limiting principle.
The pragmatist attitude is a suitable remedy against such
fallacies. It tends to gain insights concerning a large domain of
applications in which a hypothesis or a law should work, instead of
depending on intuitions spontaneously acquired (though their commonality may
induce people to take them for granted). Pragmatism claims that such
insights are crucial for advancing frontiers of science.
It has been noticed above (in §3) that the law of
gravitation was regarded as lacking a sufficient evidence, that is, as not
being duly intuitive. Such was a feeling even of Newton's himself, not only
of his opponents. Nevertheless, Newton accepted it on the basis that "it
abundantly serves to account for all the motions of celestial bodies". Now
we know that it serves to account for an astonishing number of phenomena
both in macroscale and microscale. Thus it works! And such an
efficient working must have been foreseen by Newton in a bold insight, in
spite of the lack of direct evidence.
Some limiting common intuitions were shared by greatest
thinkers, thereby delaying the dawning of ideas which were to advance the
frontiers of science. This was, for example, the case of Albert Einstein who
intuitively accepted the limiting principle that any evolution of the
universe is impossible. Following this assumption, as if it were
indubitable, he had "corrected" (in fact, corrupted) the first version of
general relativity, and restored it only after Hubble's discovery of the
expanding universe. Now we know that this restored original version of
general relativity has an enormous impact on the foundations of cosmology.
Let me mention some other examples of conflicting
intuitions, those belonging to what may me called "common sense" and those
inspiring great discoverers. Among them there is the story of the Euclid's
fifth postulate; its short and intuitive equivalant has been given by
Proclos in the form:
Given a line and a point not on the line, it is possible to
draw exactly one line through the given point parallel to the line.
For more, see:
It was Gauss who worked out the consequences of a geometry
in which more than one line can be drawn through a given point parallel to a
given line, but he did not publish this revolutionary result, because the
views of the academic circles were strongly dominated by the orthodoxy of
the limiting principle supported by the authority of Immnaunuel Kant. He had
asserted that Euclidean geometry is the inevitable necessity of
thought. Only after publishing by Niko³aj £obaczewski in 1829 and János
Bolyai in 1832 a system of geometry like that of Gauss, this discovewry came
to be known to mathematicians. However, it required a time that the new
geometry be duly appreciated, so far it was beyond the common intuition, and
this fact exercised a strong limiting impact. A full recognition followed
when non-Euclidean geometries proved to possess enormous applications in
physics, hence there appeared the acknowledgment on pragmatic grounds.
In modern physics there is a lot of paradoxical
counterintuitive statemants whose main justification consists in the fact
that they work. Let me just mention the particle-wave duality. Waves
and particles are intuitively perceived as so different categories of
entities that such a duality seems to be evidently nonsensical.
Also mathematical logic and set theory, relatively new
mathematical disciplines, happen to get limitated by certain intuitions,
some of them fairly common, other ones cultivated in some philosophical
schools. For instance, the authority of Aristotle, lasting for centuries
down, limited logic to syllogistic rules (a point firmly asserted also by
Immanuel Kant), while in the set theory the same authority inhibited the
Cantorian idea of actual infinity (Aristotle allowed potential infinity
alone). However, the modern predicate logic as well as Cantorian set theory
have gained the recognition of academic communities owing to their much
For the same reason, Gödel's incompleteness theorem
concerning arithmetic has set aside the nominalist contention that
mathematics lacks any objectual reference, and so gets limited to being a
game played with mere symbols, like chess with chess pieces. Also the
nominalistic refusal of acknowledging the existence of sets gets refuted by
the enormous efficiency of second-order logic (as discussed in Section §5).
If we try to rank this essay's key concepts according to
their significance, the first three places in such a ranking would be scored
by the notions of intuition, applications of a theory, and epistemic
necessity. The last is to denote the degree of indispensability of a
proposition, as measured with the range of its applications, theoretical as
well as technological.
In such a way, the notion of intuition gets freed from two
extremities. One of them consists in treating it suspiciously as something
esoteric that cannot be conceived in terms of sober knowledge; the other --
in treating intuition as an infallible oracle, being the cognitive authority
of the last resort (this point is conspicuous in Kant's doctrine of
synthetic a priori).
Strong and weak sides of intuition are convincingly
balanced by the economist and psychologist Daniel Kahneman. His approach has
grown highly appreciated, owing to Nobel Prize (2002), as providing a basis
to understand psychological factors of economic decisions. Kahneman's idea
is concisely rendered in the title of his book Thinking, Fast and
Slow (published by Farrar, Straus and Giroux, New York, 2011). The
slow thinking amounts to algorithmic, step by step, proceding, while
the fast one consists in flashes of intuition emerging somewhere from
the resources of subconscious memory. Such a speed and creative novelty
makes intuition indispensable for the efficiency of cognition, but does not
Failures of intuition, when they happen, are due to the
fact that intuitive perceptions result from the unconscious processing of
experiences without a critical assessments which get feasible only at the
level of full consciousness. Moreover such experiences may have a very
narrow scope, as those concerning the fall of bodies, considered above in §6;
this implies a too narrow set of consequences to be used in tests aiming at
verification. As long as one's perspective, for example in physics, does not
exceed the scope of everyday experiences alone (as was the case in antiquity
and Middle Ages), they misleadingly appear to have a high authority, being
like a certificate to act as limiting principles.
The development of instruments of research (from Galileo's
lunette up to Hubble telescope and space probes) makes it possible to
discover and measure facts inaccessible to everyday experiences. And the
creating of new mathematical theories, as Newton's calculus, enables
computation which on the basis of measurements checks reliability of
hypotheses in vast domains of applications. However, let it be noticed that
every theory overcoming old intuitions is based on some other intuitions
which remain unquestionable. E.g., the law of gravitation presupposes
intuitions of what are bodies, space, distance, multiplication, division,
Scientists happen to give up certain intuitions, even
those supported by centuries of everyday experiences, in the case of their
disagreement with a theory enjoying a wide range of theoretical and
technological applications. The pragmatist strategy does not need to be
defended with philosophical arguments, since empirical sciences in their
practice spontaneously follow such strategy in a natural and spontaneous
The same is the case in mathematical sciences, though the
awareness of this fact has less progressed so far. It was Kurt Gödel who
brought about a breakthrough in this matter (cp. §5). His leading follower
is nowadays Gregory Chaitin who after Gödel declares a perspective of
everlasting progress of mathematics. This discipline posesses the potential
to win ever new computational means due to its readiness of reforming even
own foundations, if needed for such a purpose. Here is Chaitin's statement,
much opportune to sum up the contention of this essay, especially at the
point stressed with italics by myself.
"Gödel's own belief was that in spite of his
incompleteness theorem there is in fact no limit to what mathematicians
can achieve by using their intuition and creativity instead of depending
only on logic and the axiomatic method. He believed that any important
mathematical question could eventually be settled, if necessary by adding
new fundamental principles to math, that is, new axioms or postulates.
Note however that this implies that the concept of mathematical truth
becomes something dynamic that evolves, that changes with time, as opposed
to the traditional view that mathematical truth is static and eternal."
See http://www.cs.auckland.ac.nz/CDMTCS/chaitin/charly.html, "Chaitin
interview for Simply Gödel website" (9 February 2008).
How to sum up this essay still more concisely? Let mi use
for help Ockham's famous maxim: Entia non sunt multiplicanda praeter
necessitatem. It happens to be regarded as a strongly limiting
principle, but after a reflexion it may prove to mean the opposite. An
opportunity for such reflexion comes when we try to translate the maxim into
English. What its English counterpart might be like? Since Latin grammar is
here ambiguous, the maxim can be interepreted as the following equivalence:
Entities should not be multiplied then and only then, if this is not
necessary [in order to understand the world]. "To multiply" means adding
new axioms or postulates (as told by Chaiting in the quotation above),
since in this way one introduces new objects, and so advances the frontiers
of the domain in question. Our equivalence implies the following:
- If for understanding the world it proves necessary to
multiply entities, they should be multiplied.
Again in Latin:
- Entia sunt multiplicanda, si ad mundum intelligendum id necesse est.