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TARGET ARTICLE 96

 

CATS, DUCKS AND RABBITS: SUPERPOSITION AND THE OBJECT OF PERCEPTION

by Lauri Järvilehto

15 July 2007, posted 11 August 2007

 

Abstract

[1]

In this paper I will address the problem of the object of perception in a relativistic context using the principle of quantum superposition as an analogy. It will be argued that prior to being perceived, all the possible conceptualisations of a state of affairs are instantiated in it simultaneously.  A state of affairs not perceived can thus, in effect, be said to simultaneously be all the things it can possibly be.  It is the act of perception that causes the instantiation of an individual conceptualisation in a state of affairs, and thus effectively collapses the superposition of all of its possible conceptualisations.  Thus individual objects of perception are states of affairs as conceptualised. It will also be argued that while the set of possible conceptualisations of a state of affairs is infinite, it is not arbitrary.  This entails that the world must be structured, yet in a way that cannot be exhaustively described.  A limiting case of ambiguous objects will be presented to elaborate, with a corollary to the famous “Schrödinger’s Cat” thought experiment.

 

 

Keywords

 

[2]

Relativism, constructivism, realism, ontology, perception, activity, quantum mechanics, quantum superposition, relativity.

 

 

1. Introduction

[3]

In what follows, I will address a problem that may be as old as philosophy itself.  Namely that of whether a given object of perception can be said to exist independently of the observer or not.  The debate between the realist and relativist views has considerably heated up during the 20th century, not least due to the perplexing findings in modern physics.

 

[4]

I have chosen to adopt here a relativistic starting point, which relies on radical constructivism and the relational interpretation of quantum mechanics. According to this view, we cannot conceive of an object of perception separately from the observer. An object is rather something that is conceptualised, or constructed within experience, if you will, as this thing or that by an observer. An object is a state of affairs that is conceptualised by an observer in the context of the activities he is presently participating in. This will however lead quickly to a type of anti-realism. If the object is then, effectively, a construction, what is it that we are constructing from?

 

[5]

The modern trend in relativistic philosophy seems to be to set aside ontological questions such as this as unanswerable or even irrelevant and treat the whole question as solely epistemological. I subscribe certainly to the Wittgensteinian account of the demise of “old-fashioned”, systematic metaphysics (cf. eg. Wittgenstein 2003, §6.53). I do believe, however, that if one relinquishes the demand for systematic comprehensiveness, ontological questions may still be treated as meaningful and relevant. The key here is not to look for some exhaustive, final description of the world, but rather for ways to approach the way the world is in piecemeal fashion. That is to say, we should attempt to construct various, pragmatically viable descriptions of the world, which may also extend deep into the ontological structure of the world. What I am proposing in this paper would then be one such attempt.

 

[6]

In what follows I attempt to sidestep the ontological anti-realism that seems to arise from the commitments required by a relativistic world-view. I propose here to borrow a phenomenon from quantum mechanics called quantum superposition. According to the classic interpretation of quantum mechanics, a microscopic object, such as an electron or a photon, resides, prior to being measured, as a wave containing all of its possible states. Only when the object is measured will this wave function collapse, resulting in the state measured by the observer.

 

[7]

I shall proceed to propose, as an analogy, that any non-perceived state of affairs in the world is effectively a complex wave function of all of its possible conceptualisations. Only when the state of affairs is incorporated into an activity and conceptualised, the wave function of the state of affairs will collapse resulting in its being differentiated as an individual object. An object of perception is thus the collapsed superposition of a state of affairs, which can also be described as an active relation between the observer and the state of affairs.

 

 

2. Quantum Superposition and Schrödinger’s Cat

 

[8]

Quantum mechanics is notorious for its many counter-intuitive results. Many findings of the said science seem to argue with our common sense of understanding of the world, and even to dispute the laws of logic. But despite its counter-intuitive nature, the intriguing thing about quantum mechanics is, all things considered, that it works. It is, in fact, the most effective physical theory we have of the world at the time being. Thus, while quantum mechanics may at times be very counter-intuitive, the world still seems to abide to some such rules that can be effectively described by its mathematical formulas.

 

[9]

Quantum superposition stands for the principle that prior to the measurement of the state of a particle as a variable of a given quantum system, the particle exists simultaneously in all of its possible states. Only once we measure the state of the particle does the wave function of the particle, that is to say its superposition, collapse. The measurement will then return a single result biased by the probability wave associated with the particle. Laudisa & Rovelli (2005, p. 2) describe the principle as follows:

 

[10]

The characteristic feature of quantum mechanics is that it does not allow us to assume that all variables of the system have determined values at every moment […]. When [an electron] is not interacting with an external system that can detect its position, the electron can be “spread out” over different positions.

 

[11]                                                                                                                    

This is due to the wave–particle duality of microscopic objects. As Dirac (1981, pp. 9–10) points out, all kinds of particles are associated with waves and all wave motion is associated with particles. The wave-particle duality can, for example, be observed in the so-called double-slit experiment. In the double-slit experiment light is projected through two tiny coverable holes. If only one hole is open, the photons registered by eg. a photographic plate on the other side of the holes will be distributed in a particle-like manner (as individual dots). But if both holes are open, the photons will create an interference pattern that is only possible if the photons are distributed as a wave.

 

[12]

Quantum superposition is often elaborated using the famous thought experiment presented by Erwin Schrödinger in 1935, nowadays commonly called “Schrödinger’s Cat”. Schrödinger wrote as follows:

 

[13]

A cat is penned up in a steel chamber, along with the following device (which must be secured against direct interference by the cat): in a Geiger counter there is a tiny bit of radioactive substance, so small, that perhaps in the course of the hour one of the atoms decays, but also, with equal probability, perhaps none; if it happens, the counter tube discharges and through a relay releases a hammer which shatters a small flask of hydrocyanic acid. If one has left this entire system to itself for an hour, one would say that the cat still lives if meanwhile no atom has decayed. The psi-function of the entire system would express this by having in it the living and dead cat (pardon the expression) mixed or smeared out in equal parts. (Schrödinger 1983, p. 5.)

 

[14]

The cat remains in this undead state because the wave function of the radioactive source is a combination of two states, of whom one generates a particle (that will be registered by the Geiger counter) and the other does not. The source remains in the state of superposition – that is to say, both said states are simultaneously present – represented by the wave function, Y, until observed. But as soon as somebody lifts up the lid of the box, the wave function collapses immediately because the cat will then be observed to be either dead or alive. However, prior to this act of observation, while the superposition has not yet collapsed, also the cat should, in a sense, be said to be in a superposition of both states at the same time – and thus both alive and dead simultaneously.

 

[15]

The superposition, or the Y-function, has been ontologically interpreted in a number of competing ways. According to the traditional interpretation, Y is regarded as the “real” entity, and thus all observed particles are simply specific, manifested parts of it. It has also been suggested that the observed particles could be regarded as the actual entities, and that Y would then simply be a mathematical device that predicts their localities. The latter approach, while more intuitively acceptable from the point of view of classical mechanics, results, however, in substantial logical contradictions. A third approach is the so-called relational view of quantum states, where the measured particles are seen as parts of a relationship with an observer, which would seem to avoid several problems associated with the former approaches. In the relational view, quantum systems do not have any inherent properties, but only relative properties with respect to other quantum systems. (Laudisa & Rovelli 2005, pp. 2–3.) In what follows, I shall, similarly to the latter interpretation, adopt the view that any object of perception can only be defined in relation to a perceiver.

 

 

3. The Object of Perception

 

[16]

What I mean here by an object of perception is whatever is differentiated as a single individual entity by a perceiver in a context of activities. An object is a logical simple designated with a simple name. With a context of activities I mean the sum of all the activities a human being is participating in at a given moment. With an activity I mean anything a human being does that has a specific purpose. An activity consists of acts, which means all the specific temporal changes in the human being and his immediate environment (or the human being as an immediate part of his environment) that are relevant to the activity and required for it to succeed.

 

[17]

The way we differentiate our environment depends essentially on the kinds of activities we participate in. The nature of our activities dictates the needs for the differentiation of our environment – “the patterns a cognizing organism can and does abstract from experience depend on the operations of distinction and coordination the organism can and does carry out” (von Glasersfeld 1990, p.6).

 

[18]

The ways we can interact with our environment depends on the structure of the human body. We can only perceive those things that can interact with our sense organs, and we can only directly affect those things that our extensibility can affect. Thus the human body already, in a sense, pre-contextualises the possibilities of perception. In abstracting patterns from experience, we depend also on the social convention that we participate in. When we conceptualise our environment, we need to use concepts that in turn depend on the language we use and the conceptual system determined by our social convention. This in turn results in whatever is being perceived as being perceived as sharable with other members of our community.

 

[19]

What our perception consists of is in fact the properties of the object of perception; we perceive a group of properties differentiated and delimited as an individual object. We perceive for something to be soft or hard, warm or cold, light or dark, to have this form or that, this size and that, and so forth. There is no “objectness”, per se, that we perceive, but just the sum of properties which we conceptualise as a single differentiated entity – a logical simple. Were we to remove all the properties from the object, nothing would be left at all, just like if we were to look for the “real” onion by removing its layers first.

 

[20]

The properties of an object are, in actuality, utterly relative with respect to the observer and the context of observation. If we directly observe any given property of any given object, we shall arrive quickly to see that the properties are not inherently fixed apart from being so in the conceptual scheme which we utilise to conceptualise them. Thus, the “white” wall behind my computer exhibits in fact hundreds, probably thousands of shades of grey thanks to the sunlight diffused through the foliage outside the window. The shape of the computer keyboard is not, when viewed from the angle I see it now, a proper rectangle with four right-angles, but rather a parallelogram.

 

[21]

Still, the wall is white, and the keyboard is a rectangle (these are their properties) – even when I pay attention to the fact that the immediate nature of my perception would seem to imply otherwise. This is because the wall as an object and the keyboard as an object are conceptualised as objects with the given properties. They are differentiated as individual entities and the given sets of properties are instantiated in them according to their use and relevance in the present context of activities.

 

[22]

There is no God’s view, which would define the absolutely correct perspective to the keyboard or the absolutely correct way to conceptualise the light reflected on the wall. Each object can in fact be scrutinised and conceptualised in an infinite number of ways. The only way we can ever address the objects of our perception are as objects as conceptualised. Or, following the terminology I have adopted here, as states of affairs as conceptualised.

 

[23]

The world does not consist of objects prior to being conceptualised. Because the very definition of an object is here its being a differentiated individual, and because the world can be differentiated in a practically infinite number of ways, claiming that the world itself would consist of objects is effectively reduced to nonsense. Talk of such “furniture of the world” amounts to “conceptual imperialism” – a term coined by Pihlström (1996, p. 172) in reference to the refutation of conceptual tolerance.

 

[24]

This does, of course, raise the question of anti-realism. In other words, it would seem that if there were no such objects that exist independently of conceptualisation, conceptualisations – or the perception, Berkeley-wise – would then be the only thing that properly exists. Or, at best, we should make do with positing some ambiguous “stuff” of the world, from which objects are somehow extrapolated. I do not, however, subscribe to the view that this would be the extent of ontology that can be practiced while committing to such relativistic views as presented above.

 

[25]

The world very obviously has some such structure that it can be perceived and conceptualised the way it can be. While the essence of this structure appears not to be describable, we can still construct an ontological approach by describing aspects of it. This kind of a description will, of course, be just that: a description. It will not, in other words, amount to an exhaustive exposition of the world, but it will enlighten us as to the matter of what kind of a world we live in. This kind of a description will also enable us to claim very real matters about the world. Instead of ascribing our percepts as absolute properties of the world per se, we can readily assert that from a given perception we can deduce that the world has such a property that it can be thus perceived. This kind of a relativistic approach, where properties are expressed in relation to something, needs not result in anti-realism.

 

 

4. The Superposition of States of Affairs

 

[26]

An object of perception is a conceptualisation of a state of affairs. What I mean here by a state of affairs is the state or configuration of the world prior to its being conceptualised. This concept, as used in the present paper, is rather ambiguous because it in effect refers to nothing in the sense of there being no thing to refer to. The “thingness” of things can only be talked about once states of affairs are conceptualised as objects. States of affairs cannot be said to be individual in any sense of the word as we commonly use it. In fact, it cannot even be said that the states of affairs are one or many in any meaningful way. The only thing that can be asserted about them is that when we observe an object, we observe a state of affairs conceptualised as that object.

 

[27]

A state of affairs, when not conceptualised, instantiates in it all of its possible conceptualisations. It is, in other words, simultaneously every possible thing it can be. The manifestation of a state of affairs as an object requires its conceptualisation as something. A state of affairs is conceptualised as an object when it is adopted into an activity that requires its being differentiated as an individual entity. The conceptualisation of a state of affairs is an act that is required by such an activity.

 

[28]

The same state of affairs may appear as completely different objects in different activities. States of affairs can also combine in different ways in different activities. Thus one combination of states of affairs may appear as a given object in a given activity, yet as a component of another object in another activity. This is due to the fact that what constitutes a logical simple in each given activity need not correspond to any ontological simplicity. In other words, a logical simple may very well be ontologically very complex. To be a logical simple, it suffices for the object to be differentiated as an individual within a context of activities.

 

[29]

Any given state of affairs can be conceptualised in an infinite number of ways. This is because it can always be found a new use in the context of another activity, and it can always be combined with another state of affairs. While the set of possible conceptualisations is infinite, it is, however, not arbitrary. That is to say, while a given state of affairs can be conceptualised in infinitely many possible ways, there are certain ways in which it cannot be conceptualised. Infinity does not entail arbitrariness: The structure of the state of affairs limits its possible conceptualisations.

 

[30]

For example, I cannot conceptualise my computer keyboard as a helicopter and use it to fly to Stockholm in my present context of activities. I could, of course, make up a game (and thus radically change the present context of activities) in which the keyboard would be a “helicopter”, “flying” would be waving the “helicopter” up and down, and “Stockholm” would be the kitchen. By changing the context of activities thus, the change in the relationship to the said object, locale and activity would in fact be something more than just semantic. In this context the said process could easily take place using the objects mentioned – and the nature and properties of the objects would certainly be different from what they are now. (They would, in effect, be different objects.)

 

[31]

However, what I mean in my present context of activities by “using a helicopter to fly to Stockholm” is not something that I could use my keyboard for. What is in this context (and one might add, customarily in our social convention) required for something to be a helicopter are such properties that can be described as being able to take off from ground, being able to accommodate several people, having propeller blades, and so forth, these being properties that my keyboard unfortunately lacks. And no matter how much I might change from one context of activities to another, I could not possibly use the said state of affairs (here conceptualised as the keyboard) to achieve that end which is here conceptualised as flying to Stockholm.

 


[32]

If the set of possible conceptualisations of a state of affairs would indeed be infinite and arbitrary, it would entail that the world, in essence, would be reduced to homogenic “dough” for which it would make no sense to postulate a structure of any kind. However, because the set of possible conceptualisations is in the said manner limited, it comprises a lesser infinite set than the set of all arbitrary conceptualisations. The relationship between the two is not unlike the relationship between eg. the set of irrational numbers and the set of natural numbers: Both sets are infinite, but the former has greater cardinality.

 

[33]

No exhaustive description of any state of affairs can, however, be expressed. We cannot acquire (exhaustive) knowledge of the structure of a state of affairs (and hence, of the world) due to its having an infinite amount of possible conceptualisations. Thus we can only interact with and talk about states of affairs as conceptualised as objects. From this it follows, however, that we can state of any given perceived state of affairs that it has the property of its being possible to be conceptualised as this thing or that in a given context of activities – that it must be so structured for this to be possible. And this, in turn, is something we can readily – and ontologically – assert.

 

[34]

The act of conceptualisation is relative with respect to the observer and the state of affairs. Prior to the act of conceptualisation – when “left to themselves” – each state of affairs contains simultaneously all of its possible (infinite but limited) conceptualisations. Prior to conceptualisation, a state of affairs can thus be characterised as a complex wave function of possible conceptualisations – it is simultaneously all the things it can possibly be. That is to say, it rests in superposition of all of its possible conceptualisations. This wave function of all possible conceptualisations will collapse immediately when the state of affairs is conceptualised, causing a single conceptualisation to be instantiated at that moment in that state of affairs, thus effectively causing the state of affair to appear as a differentiated, individual object. The result of the act of conceptualisation is an object of perception.

 

[35]

It should be stressed that the object of perception is not, however, a state of affairs hiding behind, say, a transcendental veil. The object is the state of affairs. The conceptualisation as an object simply means adopting a fixed relation to the state of affairs. This in turn enables us to assert of a state of affairs that it has the property of being possible to be conceived of as a given object in a given context of activities. This property is relative, yet its existence can be, context-dependently, asserted.

 

[36]

The point may further be elaborated by an analogy drawn from the theory of relativity. The concept of velocity of an object can only be expressed as relative to a fixed point of reference. Thus, no object has inherently any such property as velocity. Yet, once such a fixed point of reference is attached, it is possible to express the velocity of the object clearly with a relatively simple mathematical formula. One might actually say that prior to choosing such a point of reference, the velocity of the object is also in a kind of a superposition – that is to say, it is constantly moving in all the speeds it can move in depending on the point of reference. Yet the velocity of the object is not a “ghost” property, secondary, or anti-realistic, either. It is, after the fixation of a point of reference, a very real property of the object that can be expressed and predicted by scientific theories. Similarly, the object of perception is a real object in the context of its perception. Once the fixed point (here the observer) is established, the object is, in reality, whatever it is.

 

 

5. A Limiting Case: Ambiguous Objects

 

[37]

The act of conceptualisation is not in most cases volitional. It comes as part and parcel of a given activity. Each given activity determines in itself the instrumental objects of that activity. The conceptualisations of states of affairs cannot be separated from the activities in which they are conceptualised. This means that each state of affairs is conceptualised as an object according to its use in an activity. Each object is defined by its use, which in turn depends on the social convention we are a part of, as well as the properties of the human body. This is also the reason why we cannot arbitrarily re-conceptualise our environment to our liking.

 

[38]

We are constantly participating in a number of different activities, where various ways of differentiating our environment are required. These activities form, for each given moment, the context of activities, through which we differentiate the relevant states of affairs as individual objects. Thus we usually encounter states of affairs and combinations thereof only in their collapsed states – that is to say, conceptualised as individual objects.

 

[39]

There is, however, the limiting case of ambiguous objects. When perceiving of an ambiguous object we actually encounter superposed states of affairs. Ambiguous objects are states of affairs, which are differentiated as individual objects, but which, for some reason or the other, still rest in a superposed state. That is to say, they do not have any immediate use, but rather exhibit various possible uses. Ambiguousity may appear with certain illustrations, such as Jastrow’s duck-rabbit or Rorschach’s ink blots. Also, encountering, for example, a tool of whose use one is unsure may leave it in the superposition of its possible uses. Likewise, a person indicted of a crime is not really innocent until proven guilty, but rather in a superposed state of innocence and guilt. In effect, most objects of perception (or cognition) we entertain doubt about can be said to rest in superposition of possible conceptualisations until we decide to commit to one conceptualisation or another.

 

[40]

The peculiar nature of ambiguous objects is that their conceptualisation as fixed objects of fixed use seems to depend at least partially on the volition of the observer. Of course the decision of the interpretation of the object as this or that depends greatly on other factors as well, such as evidence favouring each possible interpretation, the hopes and expectations of the observer and so forth, but in the end, the observer can in these cases actually actively affect the definition of the object.

 

[41]

The cases of ambiguous objects illuminate the process of the perception of an object – in these cases we encounter explicitly the effect of our own conceptualisation or interpretation in an object’s being this thing or that. In some (rare) cases, one can even try out consciously switching between objects – and thus consciously collapsing the superposition of the state of affairs – such as with the duck-rabbit. Simply by deciding, one can see it at one point as a duck, and at the other as a rabbit. And one can even almost see it in its superposed, ambiguous state of a duck-rabbit, even at the risk of acquiring a head-ache while at it. (Although strictly speaking a duck-rabbit conceptualised as a duck-rabbit would in fact not be a superposed object any more; “Duck-rabbit” would naturally be an individual conceptualisation.)

 

[42]

Apart from the ambiguous objects, the states of affairs that we observe within our context of activities rest collapsed, and for those parts that they are not observed, they instantiate simultaneously all their possible conceptualisations. A stick lying in the woods is currently simultaneously a support, a weapon and a dog-tease – and an infinite number of other things as well – or, possibly, two sticks or a part of a shelter. Its objectness, or nature, cannot be defined exhaustively by listing its possible uses, or by describing its composition, because such a list would be endless. Its objectness depends solely on its use in each given activity. One might here ask, what is the stick itself then, the one that is used in all these ways? That, I would dare to argue, is a state of affairs as conceptualised in the context of the present discussion.

 

 

6. Schrödinger’s Cat Revisited

 

[43]

Before wrapping up, I shall briefly revisit Schrödinger’s cat to observe some symmetries between the present approach and the quantum principle. Schrödinger’s cat is an ambiguous object. It is differentiated as an object (a cat), but it has two mutually exclusive properties (being alive and being dead), which render it in a superposed state.

 

[44]

These two conceptualisations of the state of the cat, one being “live cat”, and the other “dead cat” are supported by evidence that is in perfect equilibrium. Thus, the cat instantiates the two properties simultaneously – it is in a superposed state as regards any outside observer. No empirical observation supports either outcome more than the other, and thus this state of superposition stays until another activity of the observer takes place.

 

[45]

There are two ways of collapsing the superposition of the cat. One is to collapse the quantum superposition by opening the lid of the box. This would in all likelihood effectively collapse the “epistemic” superposition as well, because after observing the state of the cat, it would not appear as an ambiguous object of perception anymore. Thus, the context of activities would have changed (by commencing the activity of opening the lid and ceasing the activity of waiting), and in the new context the cat would be defined perfectly unequivocally.

 

[46]

The other way of collapsing the “epistemic” superposition is by an act of will, such as was the case with the duck-rabbit. One can simply decide on behalf of one outcome or the other, perhaps relying on wishful thinking or something else. In that case, were the observer able to concretely choose one conceptualisation, the cat – as concerns the observer – would be regarded with absolute clarity as the given object until contradictory evidence would arise. In the case of the cat such an act of will would be rather unlikely – people tend to be curious enough to want to find out about such things, even if it killed the cat.

 

[47]

One can, however, imagine a number of situations, where a person would actually face the need to decide on a conceptualisation of an ambiguous object – such as in a case, where no possibility of acquiring supporting evidence in favour of one outcome or the other is to be seen, but where a position is nonetheless required. In this case the superposition would, at least for the time being, collapse by the observer’s volition.

 

 

7. Conclusion

 

[48]

The world consists of states of affairs, which can be described as complex wave functions of all their possible conceptualisations. Objects of perception are states of affairs that have been adopted into an activity and thus conceptualised. Once a state of affairs is conceptualised, its wave function of possible conceptualisations – its superposition – immediately collapses, and it appears as an individual differentiated object in that context of activities. Prior to that, however, all the possible conceptualisations are instantiated in the said state of affairs.

 

[49]

Since the state of affairs can thus be conceptualised in an infinite number of ways, it cannot be said to be any particular thing prior to being conceptualised. Being a thing requires being differentiated from other things. Thus an object, or a thing, is a state of affairs that is differentiated as an individual – a state of affairs that is conceptualised. It should be noted, though, that the state of affairs does not hide behind a transcendental veil. The object of perception is the state of affairs itself.

 

[50]

The superposition of the state of affairs usually automatically collapses when it is conceptualised in a given context of activities as a differentiated individual object. The awareness of something standing this way or that is thus mostly very clear and certain to us, because it is usually determined by the activity we are currently participating in. This in turn is dependent on social convention and the structure of the human body.

 

[51]

However, when we encounter an ambiguous object in our present context of activity, we face a state of affairs that is still superposed. This “epistemic” superposition entails that we need to actively consider the interpretation of the ambiguous object. In the case where an object cannot be immediately conceptualised as something, that given object rests in “epistemic” superposition. And, if the context of activities does not change, the superposition will only collapse if we make up our minds about it. Only then shall we perceive the given state of affairs properly as a distinct, differentiated object.

 

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REFERENCES

 

Dirac, Paul, 1981, Principles of Quantum Mechanics, Oxford University Press,          Oxford.

 

Laudisa, Frederico & Rovelli, Carlo, 2005, “Relational Quantum Mechanics”, The      Stanford Encyclopedia of Philosophy, URL = http://plato.stanford.edu/            entries/qm-relational/, downloaded March 28th May 2007, pp. 1–8. Page     numbering follows print-out.

 

Pihlström, Sami, 1996, Structuring the World. The Issue of Realism and the Nature of          Ontological Problems in Classical and Contemporary Pragmatism, Acta             Philosophica Fennica, Vol. 59.

 

Schrödinger, Erwin, 1983, ”The Present Situation in Quantum Mechanics”, Quantum             Theory  and Measurement, J.A. Wheeler and W.H. Zurek (eds.), Princeton            University Press,            New Jersey. URL = http://64.233.179.104/scholar?             hl=en&lr=&q=cache:SCXRup6qG60J:www.emr.hibu.no/lars/eng/cat/+,      downloaded May 15th 2007, pp. 1–30. Page numbering follows print-out.

 

von Glasersfeld, Ernst, 1990, ”An Exposition of Constructivism: Why Some Like It Radical”, Constructivist Views on the Teaching and Learning of Mathematics,      R. B. Davis, C. A. Maher & N. Noddings (Eds.), Reston, VA, NCTM, pp. 19– 29. URL = http://www.oikos.org/constructivism, downloaded May 12th 2007, pp. 1–12. Page numbering follows print-out.

 

Wittgenstein, Ludwig, 2003, Tractatus logico-philosophicus. Logisch-

philosophische Abhandlung, Frankfurt am Main, Suhrkamp Verlag, 1st  edition 1963, repr. 2003.

 

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[I am currently working on a philosophical dissertation concerning a systematic representation of the relationship of language and perception based on Wittgenstein and the pragmatist tradition of philosophy.  I have previously studied Wittgenstein’s Tractatus extensively.  I would characterise myself philosophically as a relativist, with a hint of what used to be holism, striving to become a realist.]

 

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Lauri Järvilehto

     e-mail <lau@kolumbus.fi>