[This article was published earlier in Alethia, issue 2:3, november 2000]

Logical Meanderings between West and East:

Aristotle, Nagarjuna and Bhaskar.

Jan Straathof

Further, if, whenever an assertion is true, its denial is false and,

whenever a denial is true, the assertion of what it denies is false,

it is not possible truthfully to assert and to deny the same thing.

Aristotle

Everything is real and is not real,

Both real and not real,

Neither real nor not real.

This is the teaching of the Buddha.

Nagarjuna

In this short piece I want to make some remarks on the differences between Western and Eastern logic, focussing

in particular on the difference between the logic of the Greek philosopher Aristotle and that of the Indian philosopher

Nagarjuna. The reason I want to address the issue of logics is because I think this approach could offer a key to the

understanding and appreciation of Roy Bhaskar' s current ontology and epistemology, as elaborated in his latest

book: From East to West.

Aristotle under siege

The two pillars on which the Aristotelian logic rest are: (i) the principium tertii exclusi or principium tertium non

datur, more familiarly known as the principle of the excluded middle or as the principle of the excluded third (PET),

and (ii) the principium contradictionis the so-called principle of non-contradiction (PNC). The latter asserts that one

cannot at the same time have both 'A' and 'not A', and the former states that one has either 'A' or 'not A', and there is

no third possibility. No one will doubt that, via these two guiding principles, Aristotle has fundamentally shaped the

subsequent centuries of development in Western philosophy and especially that of the Western sciences.

In the previous century Brouwer was one of the first mathematicians to notice that Aristotle's principles are not

always completely reliable, and especially not in those parts of the sciences that deal with categories as a 'continuum'

or as 'infinite systems'. Brouwer advocated the view that in mathematics (and maybe also in general) a consequent

deployment of the principle of the excluded third would inevitably lead to illicit ontological mathematical claims.

Brouwer therefore rejected PET and became quite suspicious of the PNC too. On Brouwer' s reading, the PET

"requires that each assumption is either correct or incorrect, mathematically formulated: that for each assumed
incorporation of systems in some way into each other, either the termination [success], or the encounter of an
impossibility can be constructed. The problem of the validity of the principium tertii exclusi is thus equivalent
to the problem of the possibility of unsolvable mathematical problems."

And straight away he offers two examples, viz., 'Is there in the decimal expansion of pi a decimal that in the long

run occurs more often than others?' and 'Are there in the decimal expansion of pi infinitely many pairs of consecutive

decimals which are equal?' (van Dalen 1999, 106-7). In more simple terms, Brouwer' s argument runs as follows: if

one has deduced an inconsistency from the proposition that, for example, there exists no number with the property E,

then one is not (yet) allowed to conclude that such a number (with the property E) really exists; such a claim would

only be legitimate if that number is really constructed. Thus according to Brouwer, to know the meaning of a mathe-

matical object is to know how to perform the practices that lead to its construction (and ­possibly- inter alia the ability

of reconstruction).

And so the question whether Pierre is or is not in the cafe is easily answered with an Aristotelian 'yes' or 'no', that is,

there is no third alternative. But ­ and this is Brouwer' s point ­ the observation of Pierre' s absence from the cafe,

will not logically allow us to conclude now that Pierre is present somewhere outside the cafe. Because it is possible

that Pierre is physically standing on the threshold of the cafe, and thus he would be neither physically inside nor outside;

and also it is possible, for example in the case where Pierre has died yesterday, that Pierre (as we know him) is actually

absent both inside and outside the cafe. The presence of Pierre can only then be asserted when Pierre is actually found

present (somewhere). This example may seem inconclusive, but let us go to the tennis court, where it may not. Here

too we will notice that it is not always easy (or even possible) to judge conclusively between a 'yes' or a 'no'. If we are

watching a tennis game, quite often it is clear that the ball is in or out, but sometimes there is uncertainty: according to

the linesman the ball was in, but the player is disputing this observation and fiercely claims that the ball was out.

Luckily with modern video equipment it is possible to take a closer look at this situation in slow motion, but even

then it is sometimes not possible to reach an Aristotelian conclusion. And even if we went some steps further and

deeper with our measurements, lets say at the quantum level, the question could still remain inconclusive. [and, by

the way, would (an Aristotelian) God possess an a priori monitor which would reveal whether the ball was really in

or out?] My speculative conclusion here is that Aristotelian logic does not rule on the tennis court. There seem to be

in principle four truth-value possibilities/alternatives, viz.:

(i) the ball is in;

(ii) the ball is out;

(iii) the ball is neither in nor out, i.e. the ball hits, so to speak, an (onto-)logical threshold, viz. that place/space that is

neither in nor out; and

(iv) the ball is both in and out, that is, per the 'quantum effect', viz. the phenomenon that an elementary particle

(electron) can be present in two places at the same time: and thus, exit PET and PNC in tennis.

Brouwer is only one of many modern critics of Aristotelian logic. Its applicability is doubted and contested across a

range of fields, from quantum mechanics to complexity theory, from world systems analysis to neuropsychology.

The main critique is directed against the exclusionism of Aristotle' s twofold logic, which could well be regarded as

a kind of Tina-logic (Bhaskar 1993, 116), in the sense that it offers only two logically acceptable possibilities: there

is no third alternative. Overall, the current state of play seems to indicate that Aristotelian logic is still quite useful

and reliable, but only in (quasi- or artificially) discrete and finite (closed) systems; when we deal with or investigate

ontological realms which lie beyond the discrete and finite, we simply don't know, and have deep suspicions!

Nagarjuna' s four-lane path

It is time to travel East and to take a look at one of the most compelling logical systems that originated there: the

logical dialectics of the buddhist Madhyamika School. Nagarjuna is regarded as the founder of this school, one of the

most important in the Buddhist tradition. The method Nagarjuna and his followers employed could be called a critical

dialectical procedure. It consists in analysing and criticizing the philosophical assumptions and arguments of opponents

by demonstrating their incoherence or internal contradictions, hence their untenability. A notorious characteristic of

Madhyamika dialectics is its purely 'negative' approach with its emphasis on Sunyata (openness, openendedness,

unboundedness, emptiness, indeterminacy, freedom, silence, space or absence (cf. McCagney 1997)) as the sole and

ultimate reality. In his analyses and critiques, Nagarjuna deployed the so called Catuskoti (fourfold) logic, more widely

known as tetralemma logic, which, it is said, was inaugurated by the first buddha Gautama himself.

According to Madhyamika tetralemma logic, every proposition could, or better ought to, be approached via four

different perspectives: (i) the proposition is true, (ii) the proposition is not true, (iii) the proposition is both true and

not true, (iv) the proposition is neither true nor not true. Construing these more ontologically, we could also render

them as the categories of: (i) neither being, (ii) nor not being, (iii) nor both being and not being, (iv) nor neither being

nor not being. There is no better way in my opinion to illuminate the deployment of this tetralemma logic than by

Nagarjuna himself, so below I give a long example from his Mulamadhyamakakarika, the so called Treatise of the

Middle Way. In Chapter 15, which analyses ­ and in the end refutes ­ the reality of Essence (Self-Nature), he writes

(Garfield 1995, 39­40):

"1.Essence arising from causes and conditions makes no sense.
If essence came from causes and conditions, then it would be fabricated.

2. How could it be appropriate for fabricated essence to come to be?
Essence itself is not artificial and does not depend on another.

3. If there is no essence, how can there be difference in entities?
The essence of difference in entities is what is called the entity of difference.

4. Without having essence or otherness-essence, how can there be entities?
If there are essences and entities, entities are established.

5. If the entity is not established, a non-entity is not established.
An entity that has become different, is a non-entity, people say.

6. Those who see essence and essential difference, and entities and non-entities,
They do not see the truth taught by the Buddha.

7. The Buddha, through knowledge of reality and unreality,
In the Discourse to Katyayana refuted both 'it is' and 'it is not' .

8. If existence were through essence, then there would be no non-existence.
A change in essence could never be tenable.

9. If there is no essence, what could become other?
If there is essence, what could become other?

10. To say 'it is' is to grasp for permanence, to say 'it is not' is to adopt the view of nihilism.
Therefore the wise person does not say 'exists' or 'does not exist'.

11. 'Whatever exists through its essence cannot be non-existent' is eternalism.
'It existed before but doesn't now' entails the error of nihilism."

This all may sound quite strange and contradictory to the ears of the Western reader, because what happens here, in

this line of argument, is that both the PET and the PNC are cancelled and violated. But before one misunderstands

this as some form of obscure scepticism or nihilism, it is important to bear in mind that the Madhyamika philosopher

is not at all rejecting the existence of the phenomenal or conventional world, s/he only claims that all phenomena in/of

the conventional world lack inherent self-nature (essence). In the final analysis Madhyamika philosophy contends,

again and again, that from whichever or from all of the four logical orientations, all propositions about ultimata and/or

essences (and the apparent ultimate essential realities they refer to) will inevitably be incoherent and untenable, and

thus the only ontological truth that seems acceptable is that all beings are devoid of self-essence: there is only one

ultimate reality ­ Sunyata.

Whatever the ontological or ethical merits of Madhyamika philosophy, the strength of its tetralemma logics lie in its

epistemic range, which offers a considerably wider set of analytical tools than Aristotle' s di(a)lemma logic. To guide

our speculations, investigations and explanations of the dialectical totality of the poly-causal, pluri-temporal, ever

changing, multicultural world-systemic reality we live in today, we seems better off with Nagarjuna than Aristotle.

When Aristotle contends that, for example, 'a cause is prior to its effect', Nagarjuna holds it possible (and even neces-

sary to presume) that, because both lack any inherent self-nature or essence, cause and effect emerged simultaneously

in a united relationship, that is, in one act/event of pratitya samutpada (dependent co-arising, or interdependent origination);

when Aristotle holds that time is essentially linear, Nagarjuna will contend that time is open/empty and certainly without

any essence such as linearity (and inter alia opening up the possibility of backwards causation); where Aristotle would

claim that the substance (ousia) of things/beings is permanent (and therefore affording the logical ground for any ceteris

paribus clause), Nagarjuna would counter with the observation that impermanence and change is ubiquitous, and further,

given the lack of any inherent essence, what substance could be granted permanence?

All in all, this Madhyamika-style reasoning may seem very abstract and too remote from everyday reality to be useful,

but it is not. I think we could easily (and fruitfully) apply this tetralemma method to the superficial dualistic mode of

conflict analysis which currently holds sway in politics and the media. Take ­ as a random example ­ the Middle East

crisis: the conflict ('who started the violence?'; 'whose holy city is this?'; 'whose forefathers are buried here?'; ' who is

making the profit?'; 'who is paying the bills?'; 'who is the master, who the slave?', and so on) is narrowed down to two

parties, the Israelis (Jews?) and the Palestinians (Muslims?), and the blame is either put on one side or on the other.

But according to the Madhyamika philosophy, as long as the conflicting parties fail to consider and analyse the further

possibility that both are to blame and at the same time neither of both are to blame, there is no way to resolution and

peace.

Meandering between Aristotle and Nagarjuna

"The fact that God is unbounded, and in principle consists (also) of infinite (layers of) depth and (zones or
swathes of) extension does not mean that he can have no positive qualities (rather he has infinite qualities) or
that he can only be defined by the via negativa, as not this, not that and so on. It does mean, however, that we
must say that God is both consciousness, love, truth, bliss and so on, and beyond consciousness, love, truth,
bliss and so on. Incidentally the relative absolute (absolute-for-us) may or may not be absolute-in-itself; and
even if it is, it may be characterisable by different degrees of unboundedness, i.e. need not be simple or
undifferentiated ...." (Bhaskar 2000, 47).

It is indeed a great surprise to read Roy Bhaskar's new book. The subject of God, only mentioned and referred to

sporadically in the earlier works, here comes explicitly to the fore and is elaborated into in what could be called

A Realist Theory of God. Whence, one wonders, this seemingly radical turn to an issue and vocabulary which only

had such a minor attention in the previous oeuvre? But my interest here is not in the 'god talk' as such, and if Bhaskar

had mentioned the word 'God' only once in this new book (substituting it by some other suitable container-words),

I would still have enjoyed it as much as I do now. Because what I am interested in here are questions like: what kind

of logic is presumed in the argument of From East to West and is it of an Aristotelian or a Nagarjunian kind, or is

it both, or neither?

The above quote is I think a nice specimen of Bhaskar's current logic. It is a meandering between Aristotelian

exclusionism (exemplified in references to 'zones or swathes', 'positive qualities' and 'absolute-in-itself'), and on the

other hand Nagarjunian inclusionism ('unboundedness', 'both ..., and beyond'). In fact, throughout the whole book

Bhaskar uses a mix of logics to prepare the way for new discourses and to discover/uncover new ontological domains

which will help to make possible further and deeper scientific explanations. Examples of the innovative concepts

Bhaskar is offering include: 'aefficacious', which opts to fuse both affective and effective powers; 'demi-being' (incl.

'demi-real being', 'demi-reality'), which denotes an ontological status that lies between being and not-being; 'ingredient

essence', which incorporates the apparent contradiction of an essence with an ingredient c.q. a non-essence; 'categorial

realism', which postulates among other things the existence of a creative intelligent (epi-)structure that is both dependent

and independent of human being; 'stratified monism', which unites two opposite ontologies, viz. that of stratification

(differentiation) and that of monism (undifferentiation); 'co-presence (co-absence)', the event or emergence of both

the present and the absent at the same time. In all these examples, whatever the other influences, the imprint of

Nagarjuna's logic is clearly visible. Aristotelian motifs are, however, also prevalent, for example in the arguments for

universal causality, dispositional realism, immanent stratification of being, ethical teleology (incl. ethical ceteris paribus),

eudaimonia, the ultimate good, the ultimate god, and so on.

Bhaskar's current logic is a bold and liberating one. (Epistemically, it is remarkably bold in deeming, among other

things, time- and astral travel, mystic experience, angelic messengers, divine whisperings, dreams, reincarnations,

'transcendental identity consciousness', 'spontaneous right action', and so on as possible media of/for knowledge

gathering). It is a logic which meanders freely between Eastern and Western thought, constantly forcing and

challenging us to broaden and deepen our readings of the various cultural traditions of the past, but above all

constructively aimed at furthering genuinely open debate and totalising understanding of the present complex

and chaotic world in which we live.

References

Bambrough, R. et al., 1963. The Philosophy of Aristotle: Introduction with translations. New York, the New

American Library.

Bhaskar, Roy 1993. Dialectic: The Pulse of Freedom. London, Verso.

Bhaskar, Roy 2000. From East to West: Odyssey of a Soul. London, Routledge.

van Dalen, D. 1999. Mystic, Geometer, and Intuitionist: The Life of L. E. J. Brouwer, Volume 1: The Dawning

Revolution. New York and Oxford, Oxford University Press.

Garfield, J. L. 1995. The Fundamental Wisdom of the Middle Way:Nagarjuna' s Mulamadhyamakakarika. New

York and Oxford, Oxford University Press.

McCagney, N. 1997. Nagarjuna and the Philosophy of Openness. New York and Oxford: Rowman & Littlefield.

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Oktober 2000 - Jan Straathof - janstr@chan.nl

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Historical introduction and Fundamental Notions

Source: Brouwer's Cambridge Lectures on Intuitionism (1951) publ. Cambridge University Press, 1981. Most of first lecture plus the appendix of fragments reproduced here.

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Brouwer’s Lectures on Intuitionism (1951)

The Transformation of Mathematical Thought

Historically, the mechanism of mathematical thought has evolved alongside philosophical shifts regarding the nature of certainty. For a long time, philosophers viewed time and space as possessing immutable properties, independent of human experience or language. Mathematics was simply the exact knowledge of these properties. Early thinkers treated familiar regularities in the experience of time and space as axioms—linguistic postulates assumed to be invariable. From these axioms, systems of properties were developed using classical logic. This era, known as the observational period, viewed logic as autonomous, with mathematics functionally dependent upon it.

However, the observational standpoint crumbled in the 19th and 20th centuries. The discovery of non-Euclidean geometries by mathematicians like Lobatchefsky, Riemann, and Einstein reduced traditional spatial science to a mere chapter within a broader, abstract science of numbers. Consequently, the "Old Formalist School" (including Russell and Hilbert) attempted to treat mathematics rigorously by stripping away elements extraneous to language. They hoped to prove that a mathematics built solely on logic and symbols would be free of contradictions. This hope remains unfulfilled, and the belief in a strictly logical proof of non-contradictory existence has largely been abandoned.

Pre-Intuitionism and New Formalism

A different approach, Pre-intuitionism (led by Poincaré and Borel), maintained a modified observational standpoint. They accepted natural numbers and complete induction as intuitively exact and independent of logic. However, they failed to find a non-linguistic origin for the "continuum" (the set of real numbers). Attempts to define the continuum via infinite sequences or logical axioms (like "completeness") without sensory evidence remained problematic. Despite these foundations, pre-intuitionists continued to apply classical logic unreservedly, even after the discovery of mathematical paradoxes cast doubt on the safety of such systems.

In response to criticism, Hilbert founded the "New Formalist School." This school postulated that while "proper" mathematics might be symbolic, "meta-mathematics"—the study of those symbols—required the intuitive grasp of natural numbers. Yet, like its predecessors, New Formalism accepted the principles of classical logic as the ultimate tool for deducing truth, leaving the existential validity of the continuum unresolved.

The First Act of Intuitionism: The Primacy of Time

Intuitionism intervenes in this history with two decisive acts. The First Act completely separates mathematics from mathematical language. It recognizes that true intuitionistic mathematics is a languageless activity of the mind, originating in the primal perception of a "move of time."

This perception occurs when a life-moment falls apart into two distinct things: one that passes away and one that is retained by memory. This separation creates a "twoity." When divested of specific quality, this twoity becomes the empty form common to all duality—the basic intuition of mathematics. By unfolding this basic intuition, one creates the system of natural numbers and the "separable" parts of mathematics. In this framework, language is merely an imperfect tool for memorizing and communicating these mental constructions; it can never guarantee truth or create new systems on its own.

The Critique of Classical Logic

Because language is secondary to mental construction, intuitionism scrutinizes the application of classical logic. While the principles of contradiction and syllogism generally hold true, the Principle of the Excluded Third (which states that every assertion is either strictly true or strictly false) is rejected for infinite systems.

In a finite system, one can judge every possible construction to see if a property holds. However, in infinite systems, this is impossible. For example, consider the assertion that the sequence "0123456789" occurs in the decimal expansion of $\pi$. Unless we find this sequence or prove it impossible, we cannot assert that the statement is true or false. Classical logic, which assumes truth exists independently of human knowledge (a view famously held by Hermite), would insist the sequence either exists or does not. Intuitionism argues that if a mathematical truth has not been constructed or proved absurd, the assertion is devoid of sense.

This leads to the concept of a Fleeing Property: a property for which we can decide if any specific number possesses it, but we have no way of calculating a number that does, nor do we know if its existence is impossible. Consequently, a mathematical assertion can exist in three states:

Proved to be true.

Proved to be absurd.

Neither proved true nor absurd, with no known algorithm to decide the matter.

The Second Act: Infinite Sequences and the Continuum

The rejection of classical logic might seem to restrict mathematics, particularly regarding calculus and analysis. One might fear that intuitionism cannot account for the continuum. This fear is addressed by the Second Act of Intuitionism, which admits two ways of creating new mathematical entities:

Admitting freely proceeding infinite sequences: Rather than requiring sequences to be predetermined by a fixed law, intuitionism allows sequences that proceed more or less freely (e.g., decimal fractions that grow without a guarantee of a final exact value).

Mathematical Species: Properties supposable for entities that satisfy conditions of equality.

By admitting freely proceeding sequences, intuitionism creates a "genuine mathematical continuum" of positive measure. This moves beyond the "linguistic continuum" of the formalists, which relied on logical axioms of completeness. This approach opens a wide field of development, allowing for a rigorous reconstruction of analysis where theorems—such as those regarding the splitting of the continuum—differ meaningfully from their classical counterparts.

Summary

Brouwer argues that mathematics derives from the fundamental intuition of time ("twoity") rather than language or external logic. By rejecting the Principle of the Excluded Third for infinite systems and accepting freely proceeding infinite sequences, intuitionism creates a distinct, constructive foundation for mathematics that avoids the paradoxes of classical formalism.

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The gradual transformation of the mechanism of mathematical thought is a consequence of the modifications which, in the course of history, have come about in the prevailing philosophical ideas, firstly concerning the origin of mathematical certainty, secondly concerning the delimitation of the object of mathematical science. In this respect we can remark that in spite of the continual trend from object to subject of the place ascribed by philosophers to time and space in the subject-object medium, the belief in the existence of immutable properties of time and space, properties independent of experience and of language, remained well-nigh intact far into the nineteenth century. To obtain exact knowledge of these properties, called mathematics, the following means were usually tried: some very familiar regularities of outer or inner experience of time and space were postulated to be invariable, either exactly, or at any rate with any attainable degree of approximation. They were called axioms and put into language. Thereupon systems of more complicated properties were developed from the linguistic substratum of the axioms by means of reasoning guided by experience, but linguistically following and using the principles of classical logic. We will call the standpoint governing this mode of thinking and working the observational standpoint, and the long period characterised by this standpoint the observational period. It considered logic as autonomous, and mathematics as (if not existentially, yet functionally) dependent on logic.

For space the observational standpoint became untenable when, in the course of the nineteenth and the beginning of the twentieth centuries, at the hand of a series of discoveries with which the names of Lobatchefsky, Bolyai, Riemann, Cayley, Klein, Hilbert, Einstein, Levi-Cività and Hahn are associated, mathematics was gradually transformed into a mere science of numbers; and when besides observational space a great number of other spaces, sometimes exclusively originating from logical speculations, with properties distinct from the traditional, but no less beautiful, had found their arithmetical realisation. Consequently the science of classical (Euclidean, three-dimensional) space had to continue its existence as a chapter without priority, on the one hand of the aforesaid (exact) science of numbers, on the other hand (as applied mathematics) of (naturally approximative) descriptive natural science.

In this process of extending the domain of geometry, an important part had been played by the logico-linguistic method, which operated on words by means of logical rules, sometimes without any guidance from experience and sometimes even starting from axioms framed independently of experience. Encouraged by this the Old Formalist School (Dedekind, Cantor, Peano, Hilbert, Russell, Zermelo, Couturat), for the purpose of a rigorous treatment of mathematics and logic (though not for the purpose of furnishing objects of investigation to these sciences), finally rejected any elements extraneous to language, thus divesting logic and mathematics of their essential difference in character, as well as of their autonomy. However, the hope originally fostered by this school that mathematical science erected according to these principles would be crowned one day with a proof of its non-contradictority was never fulfilled, and nowadays, after the logical investigations performed in the last few decades, we may assume that this hope has been relinquished universally.

Of a totally different orientation was the Pre-intuitionist School, mainly led by Poincaré, Borel and Lebesgue. These thinkers seem to have maintained a modified observational standpoint for the introduction of natural numbers, for the principle of complete induction, and for all mathematical entities springing from this source without the intervention of axioms of existence, hence for what might be called the 'separable' parts of arithmetic and of algebra. For these, even for such theorems as were deduced by means of classical logic, they postulated an existence and exactness independent of language and logic and regarded its non-contradictority as certain, even without logical proof. For the continuum, however, they seem not to have sought an origin strictly extraneous to language and logic. On some occasions they seem to have contented themselves with an ever-unfinished and ever-denumerable species of 'real numbers' generated by an ever-unfinished and ever-denumerable species of laws defining convergent infinite sequences of rational numbers. However, such an ever-unfinished and ever-denumerable species of 'real numbers' is incapable of fulfilling the mathematical function of the continuum for the simple reason that it cannot have a positive measure. On other occasions they seem to have introduced the continuum by having recourse to some logical axiom of existence, such as the 'axiom of ordinal connectedness', or the 'axiom of completeness', without either sensory or epistemological evidence. In both cases in their further development of mathematics they continued to apply classical logic, including the principium tertii exclusi, without reserve and independently of experience. This was done regardless of the fact that the noncontradictority of systems thus constructed had become doubtful by the discovery of the well-known logico-mathematical antonomies.

In point of fact, pre-intuitionism seems to have maintained on the one hand the essential difference in character between logic and mathematics, and on the other hand the autonomy of logic, and of a part of mathematics. The rest of mathematics became dependent on these two.

Meanwhile, under the pressure of well-founded criticism exerted upon old formalism, Hilbert founded the New Formalist School, which postulated existence and exactness independent of language not for proper mathematics but for meta-mathematics, which is the scientific consideration of the symbols occurring in perfected mathematical language, and of the rules of manipulation of these symbols. On this basis new formalism, in contrast to old formalism, in confesso made primordial practical use of the intuition of natural numbers and of complete induction. It is true that only for a small part of mathematics (much smaller than in pre-intuitionism) was autonomy postulated in this way. New formalism was not deterred from its procedure by the objection that between the perfection of mathematical language and the perfection of mathematics itself no clear connection could be seen.

So the situation left by formalism and pre-intuitionism can be summarised as follows: for the elementary theory of natural numbers, the principle of complete induction and more or less considerable parts of arithmetic and of algebra, exact existence, absolute reliability and non-contradictority were universally acknowledged, independently of language and without proof. As for the continuum, the question of its languageless existence was neglected, its establishment as a set of real numbers with positive measure was attempted by logical means and no proof of its non-contradictory existence appeared. For the whole of mathematics the four principles of classical logic were accepted as means of deducing exact truths.

In this situation intuitionism intervened with two acts, of which the first seems to lead to destructive and sterilising consequences, but then the second yields ample possibilities for new developments.

FIRST ACT OF INTUITIONISM

Completely separating mathematics from mathematical language and hence from the phenomena of language described by theoretical logic, recognising that intuitionistic mathematics is an essentially languageless activity of the mind having its origin in the perception of a move of time. This perception of a move of time may be described as the falling apart of a life moment into two distinct things, one of which gives way to the other, but is retained by memory. If the twoity thus born is divested of all quality, it passes into the empty form of the common substratum of all twoities. And it is this common substratum, this empty form, which is the basic intuition of mathematics.

Inner experience reveals how, by unlimited unfolding of the basic intuition, much of 'separable' mathematics can be rebuilt in a suitably modified form. In the edifice of mathematical thought thus erected, language plays no part other than that of an efficient, but never infallible or exact, technique for memorising mathematical constructions, and for communicating them to others, so that mathematical language by itself can never create new mathematical systems. But because of the highly logical character of this mathematical language the following question naturally presents itself. Suppose that, in mathematical language, trying to deal with an intuitionist mathematical operation, the figure of an application of one of the principles of classical logic is, for once, blindly formulated. Does this figure of language then accompany an actual languageless mathematical procedure in the actual mathematical system concerned?

A careful examination reveals that, briefly expressed, the answer is in the affirmative, as far as the principles of contradiction and syllogism are concerned,' if one allows for the inevitable inadequacy of language as a mode of description and communication. But with regard to the principle of the excluded third, except in special cases, the answer is in the negative, so that this principle cannot in general serve as an instrument for discovering new mathematical truths.

Indeed, if each application of the principium tertii exclusi in mathematics accompanied some actual mathematical procedure, this would mean that each mathematical assertion (i.e. an assignment of a property to a mathematical entity) could be judged, that is to say could either be proved or be reduced to absurdity.

Now every construction of a bounded finite nature in a finite mathematical system can only be attempted in a finite number of ways, and each attempt proves to be successful or abortive in a finite number of steps. We conclude that every assertion of possibility of a construction of a bounded finite nature in a finite mathematical system can be judged, so that in these circumstances applications of the Principle of the Excluded Third are legitimate.

But now let us pass to infinite systems and ask for instance if there exists a natural number n such that in the decimal expansion of pi the nth, (n+1)th, ..., (n+8)th and (n+9)th digits form a sequence 0123456789. This question, relating as it does to a so far not judgeable assertion, can be answered neither affirmatively nor negatively. But then, from the intuitionist point of view, because outside human thought there are no mathematical truths, the assertion that in the decimal expansion of pi a sequence 0123456789 either does or does not occur is devoid of sense.

The aforesaid property, suppositionally assigned to the number n, is an example of a fleeing property, by which we understand a property f, which satisfies the following three requirements:

(i) for each natural number n it can be decided whether or not n possesses the property f,

(ii) no way of calculating a natural number n possessing f is known;

(iii) the assumption that at least one natural number possesses f is not known to be an absurdity.

Obviously the fleeing nature of a property is not necessarily permanent, for a natural number possessing f might at some time be found, or the absurdity of the existence of such a natural number might at some time be proved.

...

The belief in the universal validity of the principle of the excluded third in mathematics is considered by the intuitionists as a phenomenon of the history of civilization of the same kind as the former belief in the rationality of pi, or in the rotation of the firmament about the earth. The intuitionist tries to explain the long duration of the reign of this dogma by two facts: firstly that within an arbitrarily given domain of mathematical entities the non-contradictority of the principle for a single assertion is easily recognized; secondly that in studying an extensive group of simple every-day phenomena of the exterior world, careful application of the whole of classical logic was never found to lead to error. [This means de facto that common objects and mechanisms subjected to familiar manipulations behave as if the system of states they can assume formed part of a finite discrete set, whose elements are connected by a finite number of relations.]

The mathematical activity made possible by the first act of intuitionism seems at first sight, because mathematical creation by means of logical axioms is rejected, to be confined to 'separable' mathematics, mentioned above; while, because also the principle of the excluded third is rejected, it would seem that even within 'separable' mathematics the field of activity would have to be considerably curtailed. In particular, since the continuum appears to remain outside its scope, one might fear at this stage that in intuitionism there would be no place for analysis. But this fear would have assumed that infinite sequences generated by the intuitionistic unfolding of the basic intuition would have to be fundamental sequences, i.e. predeterminate infinite sequences proceeding, like classical ones, in such a way that from the beginning the nth term is fixed for each n. Such however is not the case; on the contrary, a much woder field of development, including analysis and often exceeding the frontiers of classical mathematics, is opened by the second act of intuitionism.

SECOND ACT OF INTUITIONISM

Admitting two ways of creating new mathematical entities: firstly in the shape of more or less freely proceeding infinite sequences of mathematical entities previously acquired (so that, for decimal fractions having neither exact values, not any guarantee of ever getting exact values admitted); secondly in the shape of mathematical species, i.e. properties supposable for mathematical entities previously acquired, satisfying the condition that if they hold for a certain mathematical entity, they also hold for all mathematical entities which have been defined to be 'equal' to it, definitions of equality having to satisfy the conditions of symmetry, relfexivity and transitivity. ...

Theorems holding in intuitionistic, but not in classical, mathematics often originate from the circumstance that for mathematical entities belonging to a certain species the inculcation of a certain property imposes a special character on their way of development from the basic intuition; and that from this compulsory special character properties ensue which for classical mathematics are false. Striking examples are the modern theorems that the continuum does not split, and that a full function of the unit continuum is necessarily uniformly continuous.

Notes

Introvert science, directed at beauty, does not carry risks for consequences.

The stock of mathematical entities is a real thing, for each person, and for humanity.

The inner experience (roughly sketched):

twoity;

twoity stored and preserved aseptically by memory;

twoity giving rise to the conception of invariable unity;

twoity and unity giving rise to the conception of unity plus unity;

threeity as twoity plus unity, and the sequence of natural numbers;

mathematical systems conceived in such a way that a unity is a mathematical system and that two mathematical systems, stored and aseptically preserved by memory, apart from each other, can be added;

etc.

Fragments from a lecture 'Changes in the relation between classical logic and mathematics. (The influence of intuitionistic mathematics on logic)', given in November 1951

Classical logic presupposed that independently of human thought there is a truth, part of which is expressible by means of sentences called 'true assertions', mainly assigning certain properties to certain objects or stating that objects possessing certain properties exist or that certain phenomena behave according to certain laws. Furthermore classical logic assumed the existence of general linguistic rules allowing an automatic deduction of new true assertions from old ones, so that starting from a limited stock of 'evidently' true assertions, mainly founded on experience and called axioms, an extensive supplement to existing human knowledge would theoretically be accessible by means of linguistic operations independently of experience. Finally, using the term 'false' for the 'converse of true', classical logic assumed that in virtue of the so-called 'principle of the excluded third' each assertion, in particular each existence assertion and each assignment of a property to an object or of a behaviour to a phenomenon, is either true or false independently of human beings knowing about this falsehood or truth, so that, for example, contradictorily of falsehood would imply truth whilst an assertion a which is true if the assertion b is either true or false would be universally true. The principle holds if 'true' is replaced by 'known and registered to be true', but then this classification is variable, so that to the wording of the principle we should add 'at a certain moment'.

As long as mathematics was considered as the science of space and time, it was a beloved field of activity of this classical logic, not only in the days when space and time were believed to exist independently of human experience, but still after they had been taken for innate forms of conscious exterior human experience. There continued to reign some conviction that a mathematical assertion is either false or true, whether we know it or not, and that after the extinction of humanity mathematical truths, just as laws of nature, will survive. About half a century ago this was expressed by the great French mathematician Charles Hermite in the following words: 'Il existe, si je ne me trompe, tout un monde qui est l'ensemble des vérités mathématiques, dans lequel nous n'avons d'accés que par l'intelligence, comme existe le monde des réalités physiques; l'un et l'autre indépendant de nous, tous deux de création divine et qui ne semblent distincts quà cause de la faiblesse de notre esprit, par contre ne sont pour une pensée puissante qu'une seule et même chose, et dont la synthèse se rélève partiellement dans cette merveilleuse correspondence entre les Mathématiques abstraites d'une part, I'Astronomie, et toutes les branches de la Physique de I'autre'.

Only after mathematics had been recognized as an autonomous interior constructional activity which, although it can be applied to an exterior world, neither in its origin nor in its methods depends on an exterior world, firstly all axioms became illusory, and secondly the criterion of truth or falsehood of a mathematical assertion was confined to mathematical activity itself, without appeal to logic or to hypothetical omniscient beings. An immediate consequence was that for a mathematical assertion a the two cases of truth and falsehood, formerly exclusively admitted, were replaced by the following three:

(1) a has been proved to be true;

(2) a has been proved to be absurd;

(3) a has neither been proved to be true nor to be absurd, nor do we know a finite algorithm leading to the statement either that a is true or that a is absurd. [The case that a has neither been proved to be true nor to be absurd, but that we know a finite algorithm leading to the statement either that a is true, or that a is absurd, obviously is reducible to the first and second cases. This applies in particular to assertions of possibility of a construction of bounded finite character in a finite mathematical system, because such a construction can be attempted only in a finite number of particular ways, and each attempt proves successful or abortive in a finite number of steps.]

In contrast to the perpetual character of cases (1) and (2), an assertion of type (3) may at some time pass into another case, not only because further thinking may generate an algorithm accomplishing this passage, but also because in modern or intuitionistic mathematics, as we shall see presently, a mathematical entity is not necessarily predeterminate, and may, in its state of free growth, at some time acquire a property which it did not possess before.

See lecture above on fleeing property

One of the reasons [incorrect, the extension is an immediate consequence of the self-unfolding; so here only the utility of the extension is explained.] that led intuitionistic mathematics to this extension was the failure of classical mathematics to compose the continuum out of points without the help of logic. For, of real numbers determined by predeterminate convergent infinite sequences of rational numbers, only an ever-unfinished denumerable species can actually be generated. This ever-unfinished denumerable species being condemned never to exceed the measure zero, classical mathematics, in order to compose a continuum of positive measure out of points, has recourse to some logical process starting from at least an axiom. A rather common method of this kind is due to Hilbert who, starting from a set of properties of order and calculation, including the Archimedean property, holding for the arithmetic of the field of rational numbers, and considering successive extensions of this field and arithmetic to the extended fields and arithmetics conserving the foresaid properties, including the preceding fields and arithmetics, postulates the existence of an ultimate such extended field and arithmetic incapable of further extension, i.e. he applies the so-called axiom of completeness. From the intuitionistic point of view the continuum created in this way has a merely linguistic, and no mathematical, existence. It is only by means of the admission of freely proceeding infinite sequences that intuitionistic mathematics has succeeded to replace this linguistic continuum by a genuine mathematical continuum of positive measure, and the linguistic truths of classical analysis by genuine mathematical truths.

However, notwithstanding its rejection of classical logic as an instrument to discover mathematical truths, intuitionistic mathematics has its general introspective theory of mathematical assertions. This theory, which with some right may be called intuitionistic mathematical logic, we shall illustrate by the following remarks.

Source: Brouwer's Cambridge Lectures on Intuitionism (1951) publ. Cambridge University Press, 1981. Most of first lecture plus the appendix of fragments reproduced here.

The gradual transformation of the mechanism of mathematical thought is a consequence of the modifications which, in the course of history, have come about in the prevailing philosophical ideas, firstly concerning the origin of mathematical certainty, secondly concerning the delimitation of the object of mathematical science. In this respect we can remark that in spite of the continual trend from object to subject of the place ascribed by philosophers to time and space in the subject-object medium, the belief in the existence of immutable properties of time and space, properties independent of experience and of language, remained well-nigh intact far into the nineteenth century. To obtain exact knowledge of these properties, called mathematics, the following means were usually tried: some very familiar regularities of outer or inner experience of time and space were postulated to be invariable, either exactly, or at any rate with any attainable degree of approximation. They were called axioms and put into language. Thereupon systems of more complicated properties were developed from the linguistic substratum of the axioms by means of reasoning guided by experience, but linguistically following and using the principles of classical logic. We will call the standpoint governing this mode of thinking and working the observational standpoint, and the long period characterised by this standpoint the observational period. It considered logic as autonomous, and mathematics as (if not existentially, yet functionally) dependent on logic.

For space the observational standpoint became untenable when, in the course of the nineteenth and the beginning of the twentieth centuries, at the hand of a series of discoveries with which the names of Lobatchefsky, Bolyai, Riemann, Cayley, Klein, Hilbert, Einstein, Levi-Cività and Hahn are associated, mathematics was gradually transformed into a mere science of numbers; and when besides observational space a great number of other spaces, sometimes exclusively originating from logical speculations, with properties distinct from the traditional, but no less beautiful, had found their arithmetical realisation. Consequently the science of classical (Euclidean, three-dimensional) space had to continue its existence as a chapter without priority, on the one hand of the aforesaid (exact) science of numbers, on the other hand (as applied mathematics) of (naturally approximative) descriptive natural science.

In this process of extending the domain of geometry, an important part had been played by the logico-linguistic method, which operated on words by means of logical rules, sometimes without any guidance from experience and sometimes even starting from axioms framed independently of experience. Encouraged by this the Old Formalist School (Dedekind, Cantor, Peano, Hilbert, Russell, Zermelo, Couturat), for the purpose of a rigorous treatment of mathematics and logic (though not for the purpose of furnishing objects of investigation to these sciences), finally rejected any elements extraneous to language, thus divesting logic and mathematics of their essential difference in character, as well as of their autonomy. However, the hope originally fostered by this school that mathematical science erected according to these principles would be crowned one day with a proof of its non-contradictority was never fulfilled, and nowadays, after the logical investigations performed in the last few decades, we may assume that this hope has been relinquished universally.

Of a totally different orientation was the Pre-intuitionist School, mainly led by Poincaré, Borel and Lebesgue. These thinkers seem to have maintained a modified observational standpoint for the introduction of natural numbers, for the principle of complete induction, and for all mathematical entities springing from this source without the intervention of axioms of existence, hence for what might be called the 'separable' parts of arithmetic and of algebra. For these, even for such theorems as were deduced by means of classical logic, they postulated an existence and exactness independent of language and logic and regarded its non-contradictority as certain, even without logical proof. For the continuum, however, they seem not to have sought an origin strictly extraneous to language and logic. On some occasions they seem to have contented themselves with an ever-unfinished and ever-denumerable species of 'real numbers' generated by an ever-unfinished and ever-denumerable species of laws defining convergent infinite sequences of rational numbers. However, such an ever-unfinished and ever-denumerable species of 'real numbers' is incapable of fulfilling the mathematical function of the continuum for the simple reason that it cannot have a positive measure. On other occasions they seem to have introduced the continuum by having recourse to some logical axiom of existence, such as the 'axiom of ordinal connectedness', or the 'axiom of completeness', without either sensory or epistemological evidence. In both cases in their further development of mathematics they continued to apply classical logic, including the principium tertii exclusi, without reserve and independently of experience. This was done regardless of the fact that the noncontradictority of systems thus constructed had become doubtful by the discovery of the well-known logico-mathematical antonomies.

In a foundational controversy in twentieth-century mathematics, L. E. J. Brouwer, a supporter of intuitionism, opposed David Hilbert, the founder of formalism.

Background[edit]

The background for the controversy was set with David Hilbert's axiomatization of geometry in the late 1890s. In his biography of Kurt Gödel, John W. Dawson, Jr summarizes the result as follows:

"At issue in the sometimes bitter disputes was the relation of mathematics to logic, as well as fundamental questions of methodology, such as how quantifiers were to be construed, to what extent, if at all, nonconstructive methods were justified, and whether there were important connections to be made between syntactic and semantic notions." (Dawson 1997:48)

Dawson observes that "partisans of three principal philosophical positions took part in the debate" (ibid.) – the logicists (Gottlob Frege and Bertrand Russell), the formalists (David Hilbert and his "school" of collaborators), and the constructivists (Henri Poincaré and Hermann Weyl); within this constructivist school was the radical self-named "intuitionist" L.E.J. Brouwer.

The following sections will expand these disputes noted by Dawson.

Brief history of Brouwer and Intuitionism[edit]

Brouwer in effect founded the mathematical philosophy of intuitionism as a challenge to the then-prevailing formalism of David Hilbert and his collaborators Paul Bernays, Wilhelm Ackermann, John von Neumann and others (cf. Kleene (1952), pp. 46–59). As a variety of constructive mathematics, intuitionism is essentially a philosophy of the foundations of mathematics. It is sometimes and rather simplistically characterized by saying that its adherents refuse to use the law of excluded middle in mathematical reasoning.

In 1908:

"... Brouwer, in a paper entitled "The untrustworthiness of the principles of logic", challenged the belief that the rules of the classical logic, which have come down to us essentially from Aristotle (384–322 B.C.) have an absolute validity, independent of the subject matter to which they are applied" (Kleene (1952), p. 46).

"After completing his dissertation (1907: see Van Dalen), Brouwer made a conscious decision temporarily to keep his contentious ideas under wraps and to concentrate on demonstrating his mathematical prowess" (Davis (2000), p. 95); by 1910 he had published a number of important papers, in particular the Fixed Point Theorem. Hilbert – the formalist with whom the intuitionist Brouwer would ultimately spend years in conflict – admired the young man and helped him receive a regular academic appointment (1912) at the University of Amsterdam (Davis, p. 96). It was then that "Brouwer felt free to return to his revolutionary project which he was now calling intuitionism " (ibid.).

In the later 1920s, Brouwer became involved in a public and demeaning controversy with Hilbert over editorial policy at Mathematische Annalen, at that time a leading learned journal. He became relatively isolated; the development of intuitionism at its source was taken up by his student Arend Heyting.

In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach where mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of fundamental principles claimed to exist in an objective reality. That is, logic and mathematics are not considered analytic activities wherein deep properties of objective reality are revealed and applied but are instead considered the application of internally consistent methods used to realize more complex mental constructs, regardless of their possible independent existence in an objective reality.

Intuitionism is a program of methodological reform whose motto is that "there are no non-experienced mathematical truths" (L.E.J. Brouwer)

Brouwer, the founder of the movement, held that mathematical objects arise from the a priori forms of the volitions that inform the perception of empirical objects.^[8]

A major force behind intuitionism was L.E.J. Brouwer, who rejected the usefulness of formalized logic of any sort for mathematics. His student Arend Heyting postulated an intuitionistic logic, different from the classical Aristotelian logic; this logic does not contain the law of the excluded middle and therefore frowns upon proofs by contradiction. The axiom of choice is also rejected in most intuitionistic set theories, though in some versions it is accepted. Important work was later done by Errett Bishop, who managed to prove versions of the most important theorems in real analysis within this framework.

In intuitionism, the term "explicit construction" is not cleanly defined, and that has led to criticisms. Attempts have been made to use the concepts of Turing machine or computable function to fill this gap, leading to the claim that only questions regarding the behavior of finite algorithms are meaningful and should be investigated in mathematics. This has led to the study of the computable numbers, first introduced by Alan Turing. Not surprisingly, then, this approach to mathematics is sometimes associated with theoretical computer science.

--------------------------

L. E. J. Brouwer

Early in his career, Brouwer proved a number of theorems that were breakthroughs in the emerging field of topology. The most celebrated result was his proof of the topological invariance of dimension. Among his further results, the Brouwer fixed point theorem is also well known. Brouwer also proved the simplicial approximation theorem in the foundations of algebraic topology, which justifies the reduction to combinatorial terms, after sufficient subdivision of simplicial complexes, of the treatment of general continuous mappings.

Brouwer in effect founded the mathematical philosophy of intuitionism as an opponent to the then-prevailing formalism of David Hilbert and his collaborators Paul Bernays, Wilhelm Ackermann,John von Neumann and others (cf. Kleene (1952), p. 46–59). As a variety of constructive mathematics, intuitionism is essentially a philosophy of the foundations of mathematics.^[6] It is sometimes and rather simplistically characterized by saying that its adherents refuse to use the law of excluded middle in mathematical reasoning.

Brouwer was member of the Significs group. It formed part of the early history of semiotics—the study of symbols—around Victoria, Lady Welby in particular. The original meaning of his intuitionism probably can not be completely disentangled from the intellectual milieu of that group.

In 1905, at the age of 24, Brouwer expressed his philosophy of life in a short tract Life, Art and Mysticism described by Davis as "drenched in romantic pessimism" (Davis (2002), p. 94). Arthur Schopenhauer had a formative influence on Brouwer, not least because he insisted that all concepts be fundamentally based on sense intuitions.^[7][8][9] Brouwer then "embarked on a self-righteous campaign to reconstruct mathematical practice from the ground up so as to satisfy his philosophical convictions"; indeed his thesis advisor refused to accept his Chapter II " 'as it stands, ... all interwoven with some kind of pessimism and mystical attitude to life which is not mathematics, nor has anything to do with the foundations of mathematics' " (Davis, p. 94 quoting van Stigt, p. 41). Nevertheless, in 1908:

"... Brouwer, in a paper entitled "The untrustworthiness of the principles of logic", challenged the belief that the rules of the classical logic, which have come down to us essentially from Aristotle (384--322 B.C.) have an absolute validity, independent of the subject matter to which they are applied" (Kleene (1952), p. 46).

"After completing his dissertation (1907 - see Van Dalen), Brouwer made a conscious decision to temporarily keep his contentious ideas under wraps and to concentrate on demonstrating his mathematical prowess" (Davis (2000), p. 95); by 1910 he had published a number of important papers, in particular the Fixed Point Theorem. Hilbert—the formalist with whom the intuitionist Brouwer would ultimately spend years in conflict—admired the young man and helped him receive a regular academic appointment (1912) at the University of Amsterdam (Davis, p. 96). It was then that "Brouwer felt free to return to his revolutionary project which he was now calling intuitionism " (ibid).

He was combative for a young man. He was involved in a very public and eventually demeaning controversy in the later 1920s with Hilbert over editorial policy at Mathematische Annalen, at that time a leading learned journal. He became relatively isolated; the development of intuitionism at its source was taken up by his student Arend Heyting.

About his last years, Davis (2002) remarks:

"...he felt more and more isolated, and spent his last years under the spell of 'totally unfounded financial worries and a paranoid fear of bankruptcy, persecution and illness.' He was killed in 1966 at the age of 85, struck by a vehicle while crossing the street in front of his house." (Davis, p. 100 quoting van Stigt. p. 110.)

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Do you agree... Ten Questions about Intuitionism

This website has as its goal to investigate the opinions of the mathematical community on intuitionism. To this end it presents ten slightly provocative questions and collects answers to these questions from as many people as possible.

If you want to participate in this investigation, send your answers to the ten questions to answers@intuitionism.org. The web site also contains an annotated version of these questions which for each question presents a short explanation of how you might look at it.

Each question should be answered by yes or no or mu (= "the question is meaningless", or "it depends", or maybe, as Bas put it: "explaining how you should look at this question takes so much text that there is no point in doing so"; also use this if you do not know your answer to a question, or do not want to think about it), together with a possibly empty motivation. Try to have as little mus as possible. For each person the answers will be summarized on the main page as a string of +s, -s or #s.

Questions

Do you agree that it is impossible to define a total function from the reals to the reals which is not continuous?
Do you agree that the intermediate value theorem does not hold the way that it is normally stated?
Do you agree that there are only three infinite cardinalities?
Do you agree that the continuum hypothesis is a meaningful statement that has a definite truth value, even if we do not know what it is?
Do you agree that the axiom which states the existence of an inaccessible cardinal is a meaningful statement that has a definite truth value, even if we do not know what it is?
Do you agree that for any mathematical question it is easy to build a machine with two lights, yes and no, where the light marked yes will be on if it is true and the light marked no will be on if it is false?
Do you agree that for any two statements the first implies the second or the second implies the first?
Do you agree that a constructive proof of a theorem gives more insight than a classical proof?
Do you agree that mathematics can be done using different kinds of reasoning, and that depending on the situation different kinds of reasoning are appropriate?
Do you agree that all mathematical truths are true, but that some mathematical truths are more true than other mathematical truths?
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