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The triple nature of mathematics:
deep ideas, surface representations, formal models
Zbigniew Semadeni
Warsaw University, Poland
Introduction
At ICME-2 in Exeter, in 1972, I listened to the plenary lecture delivered by René
Thom (1972). His apt analysis clearly demonstrated the untenability of the basic
assumptions (mathematical and philosophical) of the “modern mathematics”
movement (i.e., “new math”), which had just reached its zenith before the strong
criticism expressed by Hans Freudenthal and Morris Kline. The reformers came up
with various ideas and some of the proposals were very valuable. Nevertheless, most
of the promoters of the changes kept advocating the formalist approach to school
mathematics. They overemphasized the role of set theory, of propositional calculus, of
general structures (algebraic, topological), and believed in the effectiveness of
explicitly naming the basic properties of operations (commutativity etc.) in the
computations performed by students. The implementation of their ideas resulted in
premature, unnecessary abstraction. New mathematics adherents insisted on axiomatic
systems, proofs, rigour, and precision of language. They called for abandoning the
Euclidean geometry as obsolete, and treated applications (to physics and to problems
of everyday life) as irrelevant. They also pointedly neglected meaning in mathematics
and were preoccupied with its syntax. I remember that several participants of ICME-2,
profoundly shocked by Thom's arguments against the “new math”, considered his
position as untenable (in spite of his prestige as a Fields’ medalist).
This paper* owes much to Thom’s. It concerns the epistemology of mathematics.
Let us recall that epistemology is a major branch of philosophy which may be
described as the theory of cognition, the study of the origin, nature, methods, validity
and limits of scientific knowledge. A comprehensive survey of main problems of
various epistemologies of mathematics and of mathematics education is given in
(Sierpińska and Lerman, 1996).
It should be pointed out that reflections on the nature of mathematics are
important because its image (fixed in mathematicians' minds) is conveyed to educators
and prospective teachers and then influences education (curricula, textbooks,
classroom practice) in a direct way. The failures of the “new math” have shown this in
a persuasive way. Since that time, however, the pendulum has swung back. The past
two decades have witnessed another extreme: the tendency of renouncing crucial
attributes of mathematics by some philosophers and researchers (for more on this
question, see 2.8 below).
The nature of mathematics has often been presented in terms of a duality. The
best-known of these dualities contrasts “pure mathematics” and “applied
mathematics”, and goes back to ancient times (Plato versus Archytas of Tarentum).
With a shift of emphasis, this may be presented as “mathematics as a theory” versus
“mathematics as a set of useful tools and competencies”. Towards the end of 20th
century, another contrast gained widespread popularity: “Mathematics as a body of
abstract, formal, absolute, sure, eternal knowledge” versus “mathematics as human
* This paper has been written as part of a research project supported by a grant of the Polish Committee of
Scientific Research in 2003-2006. A first version of the conception of the triad: «deep, surface, formal»
was published in (Semadeni, 2002a) and was also presented at the CIEAEM conference in Płock on July
24, 2003.
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activity, problem solving, discourse”. This list of contrasting features can be
continued. It is not uncommon to find such pairs treated, wrongly, as dichotomies
(e.g., some people draw the invalid conclusion: mathematics is a product of human
activity, therefore its theorems are as fallible as other products of human thought). In
reality, although the features of mathematics articulated in each of these three dual
pairs are naturally set in opposition to each other, we stress that they should not be
regarded as describing an either/or situation. The two components of each dual pair are
not mutually exclusive; rather, they reinforce each other. In particular, mathematics
has two faces: one pure, very abstract; the other applied, “unreasonably effective in
the natural sciences” (Wigner, 1960). In a remarkable number of instances, ideas and
theorems developed within pure mathematics (with no applications in mind) later find
unexpected, highly successful applications in science and technology. It is hard to see
how to reconcile this phenomenon with post-modernist claims about the relativist
nature of mathematics.
1. The triad: «deep, surface, formal»
1.1. The core of this paper is a proposal that the above conceptions expressed in terms
of certain dualities should be enriched by a quite different conception of the triple
nature of mathematics, namely we will argue that the nature of mathematics is better
served by distinguishing: deep ideas, surface representations and formal models of
mathematical objects*.
Two elements of this triad have their origins in psycholinguistics (the Chomsky
theory). Inspired by this theory, Thom (1972) pointed out that the domain of logic and
propositional calculus includes only the “crudest joints” of our mathematical
reasoning, representing its most superficial aspects, corresponding to the surface
structures of linguistics. These crude joints neglect the fine interactions due to sense,
which are difficult to explain or formalize. Later Richard Skemp (1982) made a
distinction between the surface structures (syntax of the mathematical symbol-system)
and the deep structures (semantics), pointing out that the meaning of a mathematical
communication lies in its deep structures. Deep structures are of key importance, but
they are mental objects which are not directly accessible to other people. Only surface
structures can be transmitted. Even within human minds the surface structures are
much more accessible. Acting on these hints, we develop the conception of the first
two elements of the triad. Nevertheless, since the word “structure” has different well-
established meanings in mathematics, we replace it by “idea” and “representation”,
respectively. Admittedly, in this way we lose an advantage offered by the word
“structure”, which connotes the structural aspects of those abstract entities. When we
deal with deep ideas and surface representations we should bear in mind their
systemic nature. Surface representations are not separate signs; they are parts of
various heterogeneous systems. Deep ideas form intricate webs, which are difficult to
analyse.
1.1.1. Surface representations of a mathematical object are signs (which can be seen,
heard, touched, manipulated) representing this object. Typically they consist of words
(spoken or written) as well as of marks and drawings on paper, blackboard, and screen
or in computer memory; however, we broaden the scope of the concept by including
gestures (expressing mathematical ideas through motion), wooden models of solids,
* We shorten the descriptions by using the auxiliary term: mathematical object. It may stand for a concept,
relation, proposition, propositional function, theorem, proof, a piece of reasoning, an algorithm, a
subroutine etc. Conceivably, any such object could be considered as an element of a suitable set.
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sets of counters (representing numbers), spatial patterns, and the like. This list includes
symbols that are subject to strict syntax rules of mathematical symbol-systems as well
as symbols that admit more flexible interpretations (examples: explaining properties of
a solid by pointing out the edges of a model; using kinetic depictions; graph-supported
visual arguments). Surface representations are essentially the same as Darstellungen
(external representations, Meissner, 2002). They serve various purposes among which
we draw attention to three:
(i) They are way of communicating mathematical thoughts, ideas, reasoning etc. to
other people. They serve as an interface between the inner world of thought and the
outer world. Each of these representations has a dual status: it is a physical thing (a
sound, a piece of chalk, a bodily movement) which can be perceived by senses and at
the same time it is a mental object, serving both as a “label” and as a “handle” with
mathematical ideas attached to it. The representation entails an interpretation of what
is perceived: it is a way of mediating between (a) the concepts/thoughts represented by
it and (b) something physical.
(ii) Surface representations are indispensable tools for working mathematically (in
computations, problem solving, proving). They somehow structure the way we
conceive of mathematics. In particular, human linguistic facility is essential for
thinking. The symbol-system of arithmetic, algebra, trigonometry, and calculus is a
powerful tool for reasoning with surface representations. Part of this surface
reasoning can be done mechanically: transformation of formal expressions leads to a
result, which can be scrutinized and interpreted in terms of the situation in question.
This may be used to produce new information from the given (e.g., the solution of an
equation, a proof of a new formula). The role of visual perception, symbol
manipulation and observation is essential in the process of transforming formulas
(which complements mental reasoning), and so is the role of habits related to details of
notation (note, e.g., respecting the difference between nx and xn). The power of
external inscriptions and of diagrammatical reasoning is stressed by Peirce (1955) and
Dörfler (2004).
(iii) Words and symbols may be names or labels of mathematical objects, and are thus
instrumental in forming abstract concepts. For instance, the word “seventeen” and the
symbol “17” are needed to separate this number from other numbers and to create a
single concept, that of number 17, while the term “Banach space” helps to form a
higher-order concept of an object of functional analysis. Moreover, the visual
similarity of certain symbolic representations may help to call attention to important
analogies.
1.1.2. Deep ideas. The construct “deep idea” cannot be defined in simple analytic
terms. At this initial stage we can only give a preliminary description:
The deep idea of a mathematical object is a well-formed abstract idea which includes
the meaning of the object, its properties, its relationships with other objects, both
mathematical and non-mathematical, in real life and physics (its “conceptual domain”,
which reflects the experience with this object), and its purposes (that is, the reasons
why this object is used and studied).
However, a deep idea is not simply a sum of such constituent parts. It has to become
mature, firm and flexible as a result of some kind of a deep mental synthesis. We
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elaborate on these points below and draw attention to some intrinsic questions
concerning the proposed construct.
1.1.3. Formal models. The triad is completed when the distinctive character of
mathematics is taken into consideration. The two elements derived from linguistics are
augmented by a third one. By a formal model of a mathematical object X we
understand the counterpart of X in an axiomatic theory. This element of the triad is the
best known and is what is generally presented in textbooks and monographs (often
without the word “model”). Therefore we discuss only its relation to the other
elements of the triad. Although we present arguments to show that deep ideas are the
most important component in the triad, formal models are an indispensable part of
theoretical mathematics, crucial for research and also for certain applications*.
Nevertheless, they can play a negative role in education if they are naively taken to
constitute shining examples of proper reasoning.
1.1.4. It should be stressed that our triad describes certain features of mathematics as a
body of present human knowledge. The theory is not meant to embrace the whole field
of human activities that may be regarded as genuinely mathematical. In particular, we
do not deal with such significant questions as the process of discovery, heuristics,
problem solving, learning new ideas, applying mathematics to problems of the real
world, even though they are crucial to the process of forming deep ideas. In other
words, we do not deal with what Freudenthal (1973, p.114; 1991, p.14) called
“mathematics as an activity”, although this is a very important aspect of mathematics
and the conception of the triad may be helpful in its study.
1.2. Examples and further comments. “Deep idea” should be regarded as a primitive
notion, explained in the context of the actual work in mathematics, by analysing
pertinent examples and specially chosen situations. Sixteen basic examples have been
selected and provided with comments that highlight significant relations between
elements of the triad, their features and limitations (the remaining, unnumbered
examples appear sporadically in various parts of the text).
FIRST EXAMPLE. The expression 9+24=33 is a surface representation of a
mathematical fact, which is represented by the symbols “9”, “2”, “4”, “3”, “+”, “=”.
The corresponding deep idea is a (broadly understood) meaning of 9+24=33 in various
contexts (in real life or mathematics), links with related statements, and possible
purposes for which this fact may be used. A formal model of the equality 9+24=33 is a
true proposition corresponding precisely to this formula, expressed in the language of
an axiomatic theory (e.g., in the Peano axiom system or in any axiom system of set
theory).
1.2.1. Further typical examples are: the deep idea of a particular concept (e.g., of a
specific number, say 24, of “negative number” in general, of “point in 3D space”,
“point in Rn ” or in a more general space, of “triangle”, “geometric figure”, “cosine”,
“derivative”, “curvature”, “stochastic independence”), the deep idea of a specific
theorem (e.g., of the theorem of Pythagoras) and of a specific proof of a theorem (to
be distinguished from the deep ideas of the general concepts: theorem and proof in
* Models should not be confused with metaphors (unless one extends the scope of the latter so as to
include everything). Although metaphors play a significant role in mathematical discourse, a formal
model of X is not a metaphor of X (analogously, an architect's model of a house is a model and not a
metaphor of a house).
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mathematics), the deep idea of a specific mathematical procedure (e.g., of solving an
equation of a given type) or a specific algorithm. We may also consider the deep ideas
of typical objects at much higher levels of abstraction, such as “quotient group”, “an
equationally definable class of algebras”, “a category” (Example 14).
In order to avoid lengthy sentences, instead of “the deep idea of an object X” we
may simply say “the deep idea X”, e.g. the deep ideas “the equality (a+b)2=a2+2ab+b2
” or “power series”. The phrase “X is a deep idea” means that it makes sense to speak
of the deep idea of X. For instance, we may say that Euler's identity eiπ = –1 is a single
deep idea; so too is the process whereby we rewrite two fractions with a common
denominator, and the algorithm for long division.
1.2.2. Deep ideas have a dual status: psychological (mental objects) and
epistemological. We distinguish between “individual deep ideas of X” in the minds of
various persons and “the deep idea of X”, which is a single abstract epistemic object,
an idealized common abstract version. The latter must have some permanent
intersubjective content, which can be analysed (Examples 1-16 show how to interpret
this). On the other hand, the individual deep ideas of X of different people need not be
identical. They are purely mental objects, invisible and inaudible (sometimes they are
not easily accessible even to their possessors). They can be communicated to other
people only by surface representations (that is, by words, symbols, drawings, gestures,
etc.) and can be theoretically, though partially, reconstructed. We may speak of an
individual deep idea of X when it is sufficiently well formed in the mind of the given
person, and this presupposes:
(α) the presence of a feeling of familiarity with the object,
(β) a sense of firm certainty that basic statements concerning X are true,
(γ) adequate understanding of X and a reasonable ability to apply the
knowledge,
(δ) robustness of understanding in cases of typical cognitive conflicts.
Hints of how to interpret the requirements (γ),(δ) (and how one can judge whether they
are satisfied) are scattered throughout this paper. It should be emphasized that
perceptually justified knowledge, mental imaging and surface-verified proofs do not
by themselves yield deep ideas.
SECOND EXAMPLE. The celebrated conception by Piaget of the so-called conservation
of (cardinal) number means that a child, at some stage of mental development,
becomes deeply convinced that the cardinality of the set consisting of, say, 10 apples
does not change when the apples are spread out so that they cover a larger area (Piaget
and Szemińska, 1941; Donaldson, 1982). When this conviction becomes stable,
context-independent and applies to any number of physical objects, conservation has
become a deep idea. This invariance of the cardinal number of moveable objects can
hardly be proved formally, because any proof would involve a mathematization of the
situation in the language of set theory, and then one would face the problem of proving
the correctness of the passage from reality to the formal model. Thus conservation is
usually not secondary, or provable, but underlies a multitude of deep ideas and
mathematizations. (Actually, we distinguish between two aspects of conservation: (i)
the conviction that the set remains the same even after being spread out, (ii) the
conviction that, if it is the same set, then it must contain the same number of elements.
However, the formation of the deep idea of conservation does not require that the
person have an explicit concept of a set.)
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THIRD EXAMPLE. The deep idea “number pi” is a single mental idea. It includes the
definition of π, the situations in which π is used, and the sense of π in various contexts
(often divorced from geometry). However, if a rigorous definition of π is required, we
have to use a theory of real numbers. We may choose, e.g., π in Cantor's theory (let us
denote this by πC), but we may just as well choose πD in Dedekind's theory. It is easy
to check that the πD is different from πC. However, this discrepancy is unimportant; πD
and πC differ formally but not substantially, and neither is privileged. It is the deep
idea of π that is used in the daily reasoning of the mathematician, who does not bother
with the remote formal models πC and πD . What counts is the deep idea of π.
FOURTH EXAMPLE. The “number four” is also a deep idea. This example shows that a
deep idea may already be well formed in the mind of a child, not necessarily gifted for
mathematics. We assume that this is the case when the child can use this number (in
the context of arithmetic operations and word problems) freely, understanding the
meaning of what he/she is doing, sensibly, flexibly. Of course, this deep idea evolves
as the child gets older, but basically it remains the same idea of “four” (in much the
same way that a growing boy remains “the same person” as he grows). Number 4 has
several formal models, which serve different purposes: the set
{Ø,{Ø},{Ø,{Ø}},{Ø,{Ø},{Ø,{Ø}}}} (in von Neumann's theory); 1+1+1+1 (in
Peano's theory); the binary representation of 4 (written, e.g., as 1002); the Dedekind
cut representation of the rational number 4.
FIFTH EXAMPLE. The deep idea “natural number” is a synthesis of its aspects: counting
number (ordinal), numerosity number (cardinal), measuring number, and more
advanced aspects. It has several formal models, the following being the best known:
1) various formalizations of the Cantor-Frege approach based on the concept of
one-to-one correspondence of elements of sets;
2) the Peano axiom system (which may be formalized without the concept of set);
3) von Neumann's definition (mentioned before) in an axiomatic set theory;
4) natural numbers defined in terms of the second-order arithmetic of real numbers.
All four of these theories differ significantly. However, a crucial criterion for the
acceptance of such a formal theory is that it must be coherent with the deep idea of
“natural number”.
1.2.3. The above examples and many other examples show that it is important not to
confuse deep ideas of mathematical objects with their formal models. Models depend
on formalizations, which are by no means unique and may even appear artificial.
Generally one should be aware that the most one can demand is that deep ideas,
surface representations and formal models correspond well to each other in limited
areas. This correspondence is only partial: there are clear exceptions to the naive
harmony one might expect between them. Some such exceptions will be discussed
below.
SIXTH EXAMPLE. Cauchy's definition “for every ε>0 there exists an N such that for all
n>N, |an – g| < ε ” (in symbols and/or words) reduces the question of what is the limit
of a sequence to some finite system of logical symbols and inequalities. A great
achievement of 19th-century mathematics was to replace a vague notion of a limit by
this clear definition and to considerably raise the standard of rigour. Put differently,
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Cauchy's definition has made it possible to replace the deep idea of a limit by a surface
representation. The price paid for this is the danger that the teaching of limits may be
reduced to formal transformations of inequalities and may give rise to the regrettable
consequence that a deep idea of limit may never be established in the student's mind.
Many teachers neglect intuition, fear that it may be misleading, and believe that it
should play no part in reasoning. Consequently, many students remember at most the
ritual incantation “for every ε >0 ...”, so that what is assumed to be formal knowledge
may degenerate into reproducing approximate versions of surface representations.
Similar remarks apply to the concept of the derivative of a function. Thurston
(1994, p.163) outlined many possible ways of thinking of the derivative (or
conceiving of it):
(1) infinitesimal (ratio of infinitesimal changes);
(2) symbolic (e.g., the derivative of sinx is cosx, the derivative of xn is nxn–1);
(3) logical (in terms of ε, δ);
(4) geometric (the slope of the tangent line, if the graph of the function has a
tangent);
(5) rate (the instantaneous speed of f (t), when t is time);
(6) approximation (the best linear approximation to the function near a point);
(7) microscopic (as if you looked under a microscope of higher and higher power).
Usually (3) is accepted as the definition; however, the deep idea includes all these
features.
1.2.4. Certain basic deep ideas (e.g., “Piaget conservation”, “transitivity of <”,
“fraction”, “polygon”, “point inside a closed curve”) are formed in a person's mind
(gradually, as a result of an extended process) before any definition is learned. But
once a sufficient basis of deep ideas has been established, the formation process for
other deep ideas can be compressed. New deep ideas (particularly in advanced topics)
can then be encountered via their definitions, by dealing with those definitions,
especially by applying them.
Now, suppose a person has already learned the definition of a mathematical
object X (in the framework of some theory) and can use it correctly. One may focus on
the following problem: Has a deep idea of X already been formed in the mind of this
person? Possible criteria seem to depend on X, on the theory, and on the context in
which such a question is asked (e.g., on what is the general knowledge and experience
of the person). In certain cases it may be helpful to ask the question: Can the person
deal with X freely as part of inner thinking, correctly, understanding what he/she is
doing, and without the need of referring to a definition of X? Of course, such a
requirement should not be interpreted mechanically; it is a clue rather than a precise
criterion.
Besides the above two possibilities (a deep idea being identified prior to any
definition, and a deep idea being identified as a result of meeting a definition), some
well-understood concepts are used by mathematicians without any definition
whatsoever - perhaps because possible definitions are artificial, or partially adequate,
or simply superfluous. The deep idea suffices.
SEVENTH EXAMPLE. The deep idea “two-dimensional rectangular array” is developed
in a person's mind by dealing both with real-life situations (tiles on a bathroom wall,
eggs in a container) and with various mathematical schemes, visualized as in the
following four examples:
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This deep idea is not based on understanding language. Sophisticated analysis of
various aspects of children's perception of geometric and arithmetical structures of
such arrays can be found in Rożek (1994). She pointed out that the concept of a
rectangular 2D-array could easily be mathematized, and outlined some formal models
of this deep idea. However, usually no such definition is explicitly stated. It is not
needed because what is actually used is the deep idea of the array. A formal definition
of a (general) double array (particularly in more complicated situations from real life)
may even obscure this concept. “When the idea is clear, the formal setup is usually
unnecessary and redundant” (Thurston, 1994, p.167).
EIGHTH EXAMPLE. The deep idea “tetrahedron” (tetrahedron is here assumed to be
closed: that is, a solid together with its boundary) corresponds to several formal
models: (a) the set of points of a tetrahedron; (b) the same set with an extra structure
consisting of its four faces, six edges and four vertices (“visible attendants” in the
sense of Hejný, 1993), in other words – a geometric complex together with its
combinatorial structure; (c) a triangular pyramid, having a structure richer than that
of a tetrahedron, with one distinguished face called “bottom”; (d) a metric space; (e) a
convex set in a vector or affine space. Such different points of view were considered
by Freudenthal (1991, p. 20) in his discussion of rich and poor structures in
mathematics. The deep idea of a tetrahedron implicitly contains the above aspects. It
cannot be reduced to (a) only. When we think of a tetrahedron, we automatically have
in mind its basic geometric features.
NINTH EXAMPLE. There is an abundance of concepts called angles (cf. Freudenthal,
1973, pp.476-494). They may be classified in several ways. In particular, angles may
be divided into two basic types. An (N)-angle (“number-angle”) is a real number
(possibly a number mod 2π or mod π) assigned to a geometric configuration (planar or
3D, oriented or non-oriented; this also includes angles between two curves or between
a curve and a surface, or between skew half-lines) or assigned to an analytically or
kinetically defined situation. An (S)-angle (“set-angle”) is a set of points (i.e., a subset
of the plane or 3D space) or a set of geometric figures. The measure of an (S)-angle is
an (N)-angle (the converse need not be true, e.g., the (N)-angles in an n-dimensional
vector space with scalar product and also angles greater than 2π that describe circular
movements do not correspond well to any sets of points). There is no easy, clear way
of translating the deep idea of a specific kind of an (S)-angle into a rigorously
formulated definition. A planar angle-region may be defined geometrically as, say,
either of the two closed regions U,W determined by an unordered pair {H,L} of half-
lines (its sides) having a common end-point v. However, this approach has weak
points. A straight angle has either no vertex and no sides or infinitely many of them.
Several formal models of the deep idea of such an (S)-angle are possible, e.g., (1) the
set U itself, (2) the same set U with an extra structure of sides formalized, say, as {U,
H, L}, (3) the set U with a distinguished vertex v, i.e., {U,v}. If an (S)-angle is defined
as an ordered pair (H,L), then it determines a single angle-region only in the case of an
oriented plane. In practice, what mathematicians use in reasoning is a (compound)
deep idea of angle. It is highly likely that many university mathematicians would not
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recall any definition, though they understand very well what they are thinking about,
and could produce an ad hoc definition of an angle that fits their deep idea.
1.2.5. We continue showing examples of various kinds of interrelations between the
three elements of the triad «deep, surface, formal», arranged so as to highlight the
complexity of these interrelations. The most important assertion of this paper is that in
case of epistemological difficulties, the deep ideas prevail over the corresponding
formal models.
TENTH EXAMPLE. The standard academic presentation of the concepts: “an ordered
pair” and “function”, based on set theory, consists of the following steps: 1°
Kuratowski's pair (a,b) is defined as {{a}, {a,b}}; 2° the Cartesian product of X and
Y is defined as the set X×Y = {(x,y): x in X, y in Y }; 3° a relation is defined as a set
of pairs (i.e., any subset of the product X×Y); 4° a function is defined as a relation
satisfying two well-known conditions; 5° a sequence (a1,..., an) is defined as a function
on {1,...,n}; 6° the product X1×...×Xn is defined as the set of sequences (x1,...,xn) such
that xj ε Xj for j in {1,...,n}.
In this example a singular cognitive conflict is hidden. Namely, the above
presentation of six well-known definitions is not so simple and neat as it looks. In fact,
it has a serious weak point: it is easy to check that an ordered pair (x1,x2) is not the
same as the sequence (x1,x2). Kuratowski and Mostowski (1952) commented: “in
applications usually it does not matter which of the two notions is used”. We rephrase
this statement saying: “In the real daily work of a mathematician only the deep idea
counts. Formal models of a pair exist and have the desired properties, but they are not
directly used”. We have a peculiar loop of concepts: functions are regarded as a
special case of relations, relations are regarded as sets of pairs, pairs are regarded as
sequences, and sequences are regarded as functions. A formal vicious circle can be
avoided (see e.g. Gödel, 1940), but it nevertheless remains in various places. For
instance, the reader of J.L.Kelley's very popular General Topology (1955) may be not
aware that the special case of the product X1×...×Xn for n=2 is not the same as the
product X1×X2 defined earlier in the same book. In this case the discrepancy is
particularly striking, because the reasoning is claimed by the author to be strictly
rigorous (in the setting of an axiomatic system of set theory). Some authors assume an
ordered pair to be a primitive notion; this, however, does not overcome the problem.
Although the five concepts involved in this cognitive conflict are part of basic set
theory, mathematicians are not troubled by this. What they actually work with is not
the formal definition but the intuitively clear deep idea of an ordered pair. The four
deep ideas: “ordered pair”, “relation”, “function”, “sequence” do not form a single
chain where each notion is defined in terms of its predecessors alone. The actual basis
of mathematical reasoning is the whole quartet of these four deep ideas, each closely
tied to the others.
Ninety years ago one of the founders of set-theoretical mathematics wrote:
“This concept [an ordered pair] is fundamental to mathematics; from a
psychological point of view, an ordered non-symmetric selective link
between two things is primal in relation to unordered, symmetric,
collective. Thinking, speaking, reading and writing are bound to
temporal succession, which suggests itself before it can be passed over.
A word is prior to the set of its letters, an ordered pair (a,b) is prior to an
unordered pair {a,b}” (Hausdorff, 1914, p. 32).
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Furthermore, the deep idea “sequence” cannot be reduced to the correspondence k →
ak. We think of (a1,..., an) as of terms a1,...,an in some order: a1 first, then a2, and so on.
Yet, no order is explicitly stated in the definition of a sequence as a function. The
order is implicit, it is induced by the natural order on {1,...,n}.
The problem of existence of two formal models: (x1, x2)pair and (x1, x2)sequence of
the deep idea “ordered pair” is markedly different from the case of two formal models
of π. The two models πC and πD (Example 3) are constructed in two distinct theories,
whereas the two definitions of X1×...×Xn are formulated in the same theory. We also
note that the case of tetrahedron (Example 8) is different from both the case of pairs
and the case of π. The choice of one of several possible formal models of a tetrahedron
depends on what structures of it are singled out.
ELEVENTH EXAMPLE. Elementary algebra appears to be based on surface
representations and formal models only. Yet, there are good reasons to believe that
algebraic deep ideas are also formed in human minds. “Each mathematician (...) will
admit that he is able to attribute meaning to each of the symbols used in each of his
proofs” (Thom, 1970). “The mathematician gives a meaning to every proposition, one
which allows him to forget the formal statement of this proposition within any existing
formalized theory (the meaning confers on the proposition an ontological status
independent of all formalization)” (Thom, 1972, p.202). In particular, algebraic
expressions and their transformations have various meanings, originating from their
role in mathematics and elsewhere. A good command of expressions such as ax2 or
a1+...+an requires the deep idea of the symbol system of school algebra: the role of
letters, indices, dots, and also distinguishing between unknowns, constants, variables,
coefficients, parameters. If a person is lacking these ideas, their verbal description
would be of little help. The emergence of the deep idea of an expression, e.g. x+4,
presupposes that the person can conceive of it as being a distinct object; the deep idea
of the expression includes its meaning, possible purposes, and its relations with other
concepts (arithmetical expressions, solving equations). The surface structure of an
algebraic expression is the arrangement of its terms and operations, e.g., x+4 and 4+x
have different surface structures, though they have the same systemic structure (i.e.,
they are equal, Kieran, 1989). Both the surface structure of x+4 and its systemic
structure are deep ideas: so, too, are the general concepts of the surface structure and
of the systemic structure of an algebraic expression (they may be formed in a person's
mind even though the person does not know such names and has never made such a
distinction explicitly).
Transforming 6x+3x or (–2x)·8x in order to “simplify the expressions” (or
“perform the given operations”) may be based on formal properties of the operations
(e.g., distributivity). However, what is more likely is the use of informal, not explicitly
specified rules, which – in the long run – may perhaps evolve into deep ideas. One
may speculate that the student procedure called automatization (Demby, 1997) is an
indication that pertinent deep ideas are being formed in his/her mind. Those students
were genuinely surprised by the question “Why do you think this is correct?” and
exclaimed, e.g.: “It’s obvious!”. A similar attitude was reported in the case of Piaget
conservation (Example 2). Many conserving children were surprised by questions
such as “Is the number of apples the same now?” (they exclaimed: “Why do you ask?
Of course it’s the same!”).
TWELFTH EXAMPLE. The deep idea of a straight line in the individual minds of Euclid,
Kant and Hilbert were certainly different, but their mathematical essence (including
e.g., the axiom “any two points lie on one and only one straight line”) is basically the
11
same (notwithstanding the inevitable philosophical and cultural differences). This
being so, we may speak of a single deep idea “Euclidean straight line” as an
epistemological object (analogously, the Nile River in the time of Euclid was different
from today's Nile, but it is still “the same river”; snowflakes are all different but there
is a single general concept “snowflake”). Similar arguments apply to many concepts,
e.g., the modern deep ideas “real number” and “the continuity of a function of real
variable” are basically the same as those in time of Weierstrass (when they matured
after a very long historical formation process) although our present knowledge about
those concepts is much richer.
1.2.6. Although the individual deep ideas of an object X in minds of specific persons
are unlikely to be identical, they always have a common core. For example, if X
denotes the expression 7(6+2), the common core includes understanding that we add
6+2=8 and then multiply 7 times 8, getting 56. If X denotes “the derivative of sine”,
the common core includes: the meaning of “derivative” and of “sine”, the fact that the
result is cosine, and the reason why that is so. Therefore, if the deep idea X in the
mind of person A and deep idea X in the mind of B are sufficiently well formed, we
regard them as epistemologically the same deep idea X. The experiences of thousands
of people over the centuries provide irefutable empirical evidence of the fact that basic
mathematical ideas are concordant in the above sense (in spite of numerous slips,
errors and changes in the past). Hence the deep idea of X may be considered as a
single abstract epistemological object. For instance, there may be individual
differences in the way people think of the number π (depending on their earlier
experience and on their knowledge), and yet everywhere in the world people who have
a good command of the concept of π and of its use must share some common know-
ledge of it (including, say, πr2 as the area of a disc), regardless of whether the
explanations are expressed in English or another language, with or without symbols.
THIRTEENTH EXAMPLE. Two separate small groups of German mathematicians
working in convexity theory were informally asked by the author whether an extreme
point (vertex) v of a convex polyhedron K in Rn is the same as a 0-dimensional face of
K (a convex subset F of K is called a face of K if the conditions xεF, x = ½y+½z, yεK,
zεK imply yεF, zεF). They answered: “Yes, of course”, and appeared not much
disturbed by the remark that such a face is a singleton {v} and not just the element v.
A possible interpretation is that they thought of a single deep idea, which has two
formal models: {v} and v. Their answers were incompatible with the set-theoretical
foundations of the theory. Although the samples were not representative, this indicates
a possible line of research. (Note that the distinction between v and {v} was strongly
emphasized by the promoters of the “new math” reforms in the 1960s, and by those
who teach elementary set theory at college level today).
FOURTEENTH EXAMPLE. The basic concepts of category theory, such as category,
functor, natural transformation of functors, are deep ideas. This makes the theory
robust when serious difficulties with set-theoretical foundations are confronted (Mac
Lane, 1971).
1.3. Proofs and proving. We shall now consider the concept of a proof, which is
central to mathematics. Proofs play many significant roles; for a comprehensive
survey, augmented with the literature of the subject, see (Hanna, 2000). The most
important functions of proofs and proving are: verification (of the truth of a
statement), justification and explanation (insight into why a statement is true).
12
1.3.1. The general concept “proof of a theorem” is a deep idea. It is often taken for
granted that the proofs found in academic books correspond well to the general
description presented in books on mathematical logic. A proof of a theorem T in a
given formalized axiomatic theory is (loosely speaking) a sequence of propositions
T(1),...,T(n) (expressed in the language of the theory) such that T(n) is just T and each
T(k) can be deduced from the axioms and the preceding propositions T(1),...,T(k–1) by
using one operation from a given list of admissible ways of inference. However, no
research proofs (except for certain publications in logic) are written in this way, which
is practically unrealizable. This question is discussed in some detail in (Mac Lane,
1981) and (Davis and Hersh, 1981). It may be summarized by the following quotation
from Mac Lane (in: Atiyah et al., 1994, p. 191):
“The sequence for the understanding of mathematics may be: intuition,
trial, error, speculation, conjecture, proof. The mixture and the sequence
of these events differ widely in different domains, but there is general
agreement that the end product is rigorous proof – which we know and
recognize, without the formal advice of the logicians”.
The words after the dash may be interpreted as follows: proof is a deep idea which is
formed in a long process as a result of hard work with mathematics; the way logicians
speak of proofs is valuable, but is not helpful when it is necessary to write down or
verify a difficult proof (and is completely useless when a proof has not yet been
conceived). Thus, formal models of proofs of typical theorems exist potentially, in
highly idealized versions, but are not actually executed. In case of proofs, the distance
between “deep” and “formal” seems to be greater than that in the previously
considered examples.
“We should recognize that the humanly understandable and humanly
checkable proofs that we actually do are what is most important to us,
and that they are quite different from formal proofs. For the present,
formal proofs are out of reach and mostly irrelevant” (Thurston, 1994, p.
171).
FIFTEENTH EXAMPLE. In one of the school textbooks in the 1960’s, the author
introduced vectors and the scalar product, believing that these important concepts
should offer some advantages. Indeed, they were used, in particular, to give a very
short and elegant proof of the theorem of Pythagoras. Specifically, if vectors a and b
forming two sides of a triangle are perpendicular, then |a–b| is the length of the
hypotenuse, which can easily be computed: |a–b|2 = (a–b)(a–b) = |a|2 – 2ab + |b|2.
Since the scalar product ab is 0, the theorem follows immediately. Nevertheless, some
top students insisted later that no proof of the theorem of Pythagoras was given,
although they remembered the computation. They could check the surface part, but the
passage from the above equalities to the conclusion required deep ideas which were
lacking in case of those students.
1.3.2. Actual proofs combine reasoning based on deep ideas (D) with making use of
surface representations (S). Extensive use of (S) may have an adverse effect: “The
more easily recorded and communicated formalism and machinery tend to gradually
take over from other modes of thinking” (Thurston, 1994, p.167). Clearly, each proof
must involve (S). This is even true when the proof is just an oral explanation that
13
makes no use of symbols; still, it uses words, and words are (S). When someone edits
or verifies a proof and wants to understand why the successive steps are valid, (D) is
also involved. A single step in a research proof may turn out as a “big jump” if
compared with single steps in a formalized theory. On the other hand, a frequently
applied and verified sequence of typical steps may become a kind of “subroutine”, a
new “obvious step”, and eventually a new deep idea. Only proofs of very special
theorems are free of deep ideas. Thus, almost all proofs involve both (D) and (S). The
relations between the roles of (D) and (S) in proofs and the question which of them is
dominant have been subject of many casual remarks as well as of research. A
celebrated characteristic was given by Poincaré (1908, p.133): “When a logician
decomposes a proof into many elementary steps, each correct, he will not yet have the
whole; this indefinable something that endows the proof with unity will escape the
net”. This may be rephrased by distinguishing (i) step by step reasoning and (ii)
comprehensive reasoning. In a single step, (S) is indispensable and (D) is usually also
needed, although such steps may often be reduced to a formal application of rules,
e.g., in a proof of an algebraic formula or in a geometric proof which consists in
finding a chain of congruent triangles. On the other hand, with the exception of trivial
cases, full understanding of the proof requires deep ideas.
SIXTEENTH EXAMPLE. In an academic textbook (Sierpiński, 1951, p.98), written by one
of most famous Polish mathematicians, the proof of the fundamental theorem of
algebra is based on the following lemma: If f is a non-constant polynomial with
complex coefficients and f(z0) ≠ 0, then there exists a complex number z1 such that
|f(z1)| < |f(z0)|. The theorem follows by noting that the function |f(z)| is continuous and
must attain its minimal value at some z0. If f(z0) were different from 0, this would
contradict the lemma. I vividly remember reading this when I was a student. I
understood each of some twenty steps of the proof of the lemma based on elementary
properties of complex numbers, but remembering the two pages of such computations
seemed hopeless. Some thirty years later I suddenly realized that the lemma was so
easy that an oral proof would do. Indeed, without loss of generality we may assume
that z0 = 0 and f(0) = 1. Suppose first that f is a polynomial of degree 1, i.e., f(z) =
1+az. Let z = r(cosφ+isinφ) and let φ change from 0 to 2π. If r is fixed, then az
revolves around the point 0, and 1+az revolves around 1. If r is small enough, then for
some z1 the point 1+az1 must be closer to 0 than the point 1. If f(z) = 1+az+a2z2+... then
for sufficiently small r the higher powers akzk are so small that they cannot compensate
the distance resulting from 1+az. In the case where a1 and f(z) = 1+apzp+... (p>1) the
argument is similar, 1+apzp revolves p times. I looked again at the proof in the book: it
used essentially the same argument as my “oral proof ”, but without reduction to easier
cases and without any reference to geometry (still, it gave an explicit construction for
z1).
Sierpiński's proof – a typical manifestation of the attitude prevailing in the 1950s
– is a vivid example of what Lakatos (1976, p. 142) called the Euclidean deductivist
style. There is no hint why the argument works; surface representations dominate and
the role of deep ideas is minimized. On the other hand, the above “oral version” is
based on deep ideas and provides nervus probandi (the crucial idea of the proof which
makes it valid), consisting of two observations: (i) in a neighbourhood of z0 the
general polynomial f behaves approximately as a polynomial of the form 1+apzp; (ii) if
f(z) = 1+apzp, then the lemma follows immediately from the geometric interpretation of
1+apzp on a circle around z0. Sierpiński's proof is oriented towards demonstrating the
truth while the aim of the “oral version” is twofold: justification as well as better
understanding why the lemma is true.
14
1.3.3. Several authors have discussed distinctions of this kind. M.Steiner differentiates
proofs that explain from proofs that only demonstrate. Wittmann and Müller elaborate
“content-insight proofs” which focus on the meaning. Simpson highlights “proofs
through logic”, which emphasize the formal, and “proofs through reasoning”, which
involve investigations and heuristics; for references and further details in this line, see
(Hanna, 2000). Analogously, Weber and Alcock (2004) distinguish a syntactic proof
production from a semantic proof production. All these differentiations are related to
those between (S) and (D) outlined in 1.3.2.
Raman (2003) distinguishes between private argument (which engenders under-
standing) and public argument (with sufficient rigour for the mathematical
community). She also speaks of three types of ideas used in producing a proof. The
first is called a “heuristic idea”; it is essentially private and gives a sense of
understanding and a feeling that the statement ought to be true, but not conviction. The
second type, called a “procedural idea”, is essentially public and is based on logic and
formal manipulations, which lead to a formal proof; it gives a sense of conviction, but
not understanding. The third, called a “key idea”, is a link between a heuristic idea and
a procedural one, a “mapping” of the first to the second. If somebody has a key idea of
a proof, he/she is able to see that both the heuristic idea and the procedural one
represent the same idea. Example 16 fits this conception perfectly; the “oral version”
is a key idea and a private argument acceptable to a person with sufficiently formed
relevant deep ideas that are involved in it, but for a wider audience some details must
be elaborated. The same example shows that the contrasting pairs: “demonstrate-
explain”, “public-private”, “procedural-heuristic” need to be augmented with the pair
“surface-deep”.
2. Relations of the conception of the triad to other theories
2.1. Formation of deep ideas in the human mind. The conception of individual deep
ideas is based on the assumption that they are constructed in the minds of people.
Psychological theories like those of Piaget (Beth and Piaget, 1966; Piaget, 1971;
Piaget and Inhelder, 1989; Piaget and Garcia, 1989), van Hiele (cf. Freudenthal, 1973,
pp. 407-416; Tall and Thomas, 2002, pp.27-47) and other authors provide some
insight into the multi-step process of the formation of deep ideas at various levels of
cognitive development. An operation on deep ideas (such as “numbers”, “isometries”,
etc.) may later become a deep idea at the next level, and the process is repeated.
Structures are constructed which are later structured by new structures. In particular,
certain deep ideas are possible only when a suitable level of reifications of actions into
entities (Kaput, 1989; Sfard, 1991) or encapsulation (Dubinsky, 1991) has already
been attained; the deep idea of 9+24 = 33 cannot emerge before the person is able to
grasp such equalities proceptually in the sense of Gray and Tall (1994); see also Gray
in (Tall and Thomas, 2002, pp. 205–217).
2.2. Intuition. Certain features of deep ideas bring them close to other familiar
constructs. We first discuss the intuition. Mathematicians are too ready to invoke
inner intuition when no other ground of knowledge can be produced (Frege, 1884).
However, in the context of mathematics and mathematics education the word
“intuition” in used in many markedly different senses. Apart from the interpretation
attributed to Descartes or Kant (non-inferential knowledge, the direct knowing without
the conscious use of reasoning), several other ways of using the word by
mathematicians are vividly described in (Davis and Hersh, 1981). Substantially
15
different features of intuition are considered in (Fischbein, 1987); see also (Otte,
1994). In (Kitcher, 1983) intuition is presented as one of most overworked terms in the
philosophy of mathematics. My position is the following: Although certain deep ideas
in certain situations may be referred to as “intuition”, the difference between the two
concepts is essential, since deep ideas stem from conscious mathematical activities
and from reasoning (also, a deep idea is not “a specific mathematical intuition that is
the genetic origin of concepts prior to experience”). Therefore it is best to separate the
clear conception of a deep idea from the many confusing usages of the word
“intuition”. Moreover, the popular stereotypical image “formal mathematics versus
intuition” is a false dichotomy, a dichotomy that results from distorted perspective.
2.3. Meaning. Following the arguments of Thom (1970, 1972) and Skemp (1982) we
may try to describe the essence of the deep idea of an object X as the meaning of X.
Then, however, the question arises of how to describe what is the “meaning”.
Mathematicians either regard the word “meaning” as a non-technical informal word of
ordinary language or reduce it to some definiens, e.g., by saying that the meaning of
“square” is “rectangle with equal sides” (this property is, of course, part of the deep
idea “square”, which embraces much more). It is well known that no satisfactory
theory of “meaning” or “sense” can be found in texts on philosophy, logic, semiotics
or linguistics (Quine, 1980, Essays II, III, VII; Sierpińska, 1994). If the meaning of an
expression is understood as “the idea expressed by it” (Quine, 1980, Essay III), then
explaining deep ideas by meaning is getting us nowhere. According to various
theories, a mathematical object X may have a well-determined pre-existing meaning
that we study and describe, or the meaning of X is constructed by us in our minds, or
the meaning of X is a certain way of understanding X. Let us also note that according
to Cobuild English Dictionary, “the meaning of a word, expression or gesture is the
thing or idea that it refers to or represents and which can be explained using other
words”. However, in many cases the meaning of a mathematical object (as we
interpret it) cannot be fully explained by words.
2.4. Concept. One may argue that the deep idea of an object X (such as, say, √2) is the
same as the concept of X, and hence the term “deep idea” is superfluous. The deep
idea of √2 includes the concept of √2, indeed, but the differences between them are
essential. A person may acquire the concept of √2, that is, may understand its
definition and be able to use √2 correctly even if the deep idea of √2 has not been
formed (as judged on conditions (α), (β), (γ), (δ) in 1.2.2 above). Moreover, we need a
separate term because in the literature the term “concept” is assigned various
incompatible senses. Examples: in logic, concept may be “the meaning of a name”
(Ajdukiewicz, 1974; Sierpińska, 1994, p.42). In philosophy, concept may be “a word
which has a general meaning; knowledge of a concept is what enables to define a
word” (Vesey and Foulkes, 1990); other descriptions use the Aristotelian conception
of essential properties that characterize the designati of the concept. According to
Freudenthal (1991, p.18),
“Concept of X seems to mean how one conceives of an object X in a
certain perspective, say, by inspection, reflection, analysis, scrutiny, or
whichever you wish”, “What is the difference between number and
number concept (...), between X (an object) and the concept of X? (...)
There is at any rate a difference between both of them.”.
16
His statements would please Platonists, since they connote a distinction between an
object (e.g., number) and the concept of it in human minds, which makes this object
external to human thought. Mathematicians usully interpret “concept X” as it is stated
in the definition. This is precise. In contrast, “deep idea X” is more comprehensive. To
complete the picture, we note that the word “concept” is never used in (Beth and
Piaget, 1966) except for quotations from other authors.
2.5. Mental images. Individual deep ideas have certain features of mental images (in
the sense of Tall and Vinner, 1981), of mental objects (Freudenthal, 1991, p.18), of
personal modifiable internal representations (Goldin, 2002) and of Vorstellungen
(Meissner, 2002). Although generally the individual deep idea of X includes the
mental image of X, a person may have a transient private mental image of say, logx,
whereas the understanding of it may be inadequate or not robust; in such a case, logx
is not an individual deep idea.
2.6. Instrumental and relational understanding. Skemp distinguished between instru-
mental understanding (choosing and applying rules without knowing why) and
relational understanding (knowing both: what to do and why); see (Tall and Thomas,
2002). The former means restricting the task to surface representations while the latter
either involves relevant deep ideas or paves the way for the emergence of deep ideas
in the future.
2.7. Relations to classical philosophies of mathematics. We want the proposed theory
to be philosophically neutral as much as possible. The conception of deep ideas does
not require a definite ontological commitment. It is compatible with some forms of
Platonism, in particular with “methodological Platonism” (Mac Lane, 1986, p.447),
with moderate formalism (reduced to formal models) and with our tenet that
mathematical knowledge cannot be simply transferred ready-made from the teacher to
the learner and has to be actively built by the latter in his/her own mind. We should
draw, however, the reader’s attention to the groundlessness of certain inferences. Our
tenet does not imply that knowledge is independent of the external world. It does not
imply that knowledge does not reflect certain timeless regularities of the world. It does
not imply the impossibility of objective truth in mathematics (relative, of course, to
axiomatic systems; cf. Goldin, 2003). We do not posit the primacy of the mental over
the external. On the other hand, Platonism (meant as existence of mind-independent
abstract objects whose properties humans attempt to discover and/or describe) does
not imply that numbers must be identified as sets (i.e., identified with their formal
models). It does not imply that discovery learning and group learning are impossible.
Deep ideas can be so familiar and natural to their possessor that they engender a
belief of their necessity and of objective existence; this makes Platonism plausible to
mathematicians. To forestall a misinterpretation, we emphasize that the conception of
deep ideas is equally valid with and without Platonism, but is hardly reconcilable with
nominalism, logicism, apsychologistic conceptualism, intuitionism, and radical
apriorism. Moreover, if Platonist objects exist independently of human mental
activity, mathematicians do not access them by some special “intuition”, but by
mentally constructing their isomorphic (or perhaps homomorphic) images –
interpreted as deep ideas. We regard mathematical knowledge as the heritage of
generations of creative scientists, sustained by community approval, disseminated by
accepted authorities, retraced and partially reshaped by followers (Kitcher, 1983, and
his “evolutionary epistemology”; Davis and Hersh, 1981).
17
2.8. Relations to new philosophies of mathematics. The conception of deep ideas is
compatible with moderate constructivism and with moderate social constructivism, but
it is not compatible with radical constructivism, radical social constructivism, and –
generally – with those theories that are a priori dismissive, are based on denying the
very foundations of rival theories, with radical ultrarelativist “isms” that deny the very
possibility of objecttive truth, knowledge and validity (hence their statements are
nonfalsifiable): The truth of a theorem is a “social construct” or a “social consensus”,
subject to negotiation and change, and the negation of a theorem is, possibly, “an
alternative viewpoint”. (e.g., post-modernists, radical constructivists) who view
mathematics as if it were a social science. There are only conceptions and no
misconceptions. For criticism of such positions and references, see (Freudenthal,
1991, p.142-147; Goldin, 2002; Goldin, 2003; cf. Freudenthal, 1973, p. 89). If such a
theoretical “paradigm” is unable to provide an adequate explanation of some funda-
mental question, then its protagonists simply evade the problem by declaring that it is
irrelevant or that it is only a matter of language, of social convention, etc. Claims of
this sort attract favour of those educators who oppose the closed-minded, “absolutist”
views of mathematicians (they also match the bitter experience of those myriads of
people who took behaviourist courses in mathematics and were never able to
remember and correctly use the tangle of rules imposed by teachers). On the other
hand, some of new philosophical ideas have a sound core and support positive, highly
significant educational changes, e.g., an attitude of tolerance and openness towards
visually-oriented activities, discovery process, open-ended problem solving, students'
thinking and conceptions, real-life context-embedded learning. A vital question is at
what point this openness becomes permissiveness and tolerance of serious errors. The
line between them is thin. The research paradigms of mathematics education seem to
be drifting away from the paradigm of mathematics itself (Sfard, 1998). Moreover,
there is a regrettable and widening gulf between the philosophy of research
mathematicians (respectively, scientists) and the philosophy of philosophers and
educationalists dealing with mathematics (respectively, natural sciences). The long-
term effect of these trends on education is to be judged by the next generation.
3. Concluding points
● The central tenet of the proposed theory (its hard core in the sense of Lakatos, 1978)
is that the triad «deep, surface, formal» provides an adequate framework to
discussions of the nature of mathematics as a body of knowledge. Formal logic alone
is not sufficient to explain some very basic facts concerning mathematical reasoning
(even if the analysis is confined to the final product of it, available in a published
form), because it requires a prior setting of a precise admissible language. Therefore
formalized reasoning is restricted to surface representations only and (as shown
above) it cannot fully explain some fundamental features of mathematical objects.
However, formal models are important as tools of global organization of mathematics,
and are indispensable in the case of more advanced deep ideas, ideas which cannot be
simply abstracted from activities involving real-life objects. Surface representations
are not only means to communicate mathematics, but are also invaluable tools for
mathematical reasoning and computations, and are instrumental in forming and
developing concepts.
● The features of deep ideas are described in the present paper in the context of
pertinent examples. Most of mathematical reasoning is controlled by deep ideas,
which prevail over the corresponding formal models in case of a cognitive conflict.
18
Deep ideas originate from conscious mathematical activities and from reasoning in
situations arising in real life, science, and mathematics itself. They form a complex
web of concepts linked by a whole host of types of meaning-based relationships
(which depend on a wide variety of types of activities of their origins), described only
partially in the literature (for a study of relationships such as “the same” and “can be
identified” see Semadeni, 2002b). In the process of historical development, after
having reached a certain level of maturity, deep ideas keep their identity. In spite of (i)
differences between the individual deep ideas and (ii) the changes due to the evolution
of mathematics, there is a common core in any deep idea once it has sufficiently
matured; there is no essential ambiguity concerning its basic properties. Languages are
diverse, but individual deep ideas are concordant. This is why mathematics is
universal.
● Hopefully, the conception of deep ideas may act as a bridge between the Platonist
attitude of mathematicians and the constructivist trends among researchers in
mathematics education, and hence it may help to reconcile these divergent positions,
easing the problems mentioned in 2.8.
E-mail address: semadeni@mimuw.edu.pl
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