Posts Tagged ‘Philosophy’

This problem has to do with whether a certain class of hard-to-solve problems, designated NP-Complete, can be reduced to fast-to-solve problems, designated P.   Both NP-Complete and P problems belong to NP, a set of problems for which any given solution can be verified quickly.

The time it takes to find a solution for a problem varies for reasons beside the fact that they are dealing with differently sized inputs.  The set of P problems are known to be solved within a time proportional to the size the input; this makes them ‘fast-to-solve’. On the other hand, NP-Complete are characterized by the fact that the fastest known solutions to them take time proportional to an exponential function. In solving them, we usually can’t be more clever than by checking every possible solution in sequence. Thus the time required to solve these problem increases very quickly as the size of the input grows.  It grows so quickly, in fact, that solving a problem with a moderately large input can take the billions of years, using any amount of computing power available today.

A property of NP-Complete problems is that all problems within the set can be transformed to all the other problems within the set, making the problems in NP-Complete essentially reworded versions of each other. Thus if we could find a way to solve one problem quickly (i.e. reduce it to a P problem), we can solve them all quickly (and show P=NP).

A reason for the belief that P ≠ NP is that after 30 years of research, no one has been able to find a polynomial-time algorithm for any of more than 3000 important known NP-complete problems.

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The purpose of the assignment was to explain the problem in general terms and explain its importance for the general public.

After receiving a poor grade on this, I followed up with the following email:

“Hi,

I know my explanation for the belief that P doesn’t equal NP is terse and I know I could have elaborated more but I think that ultimately, it really is the only answer. To take one example, consider cryptography. Billions of dollars worth of assets are protected by encryption protocols that are in essence NP-Complete problems. Because P!=NP is not proven, inductive reasoning (i.e. no known solutions to NP-Complete problems exist) and intuition (it doesn’t ‘feel’ like there will be solution) are the only reasons why so much trust is placed on those protocols. Further, NP-Complete problems are found across the entire spectrum of math and science. The fact that so many smart people from disparate fields of study (NP-Complete problems are found in physics, chemistry, biology, cryptography, mathematics, computer science) could not find one solution since the problem was formulated three decades ago is what ultimately lets a multi-national bank trust its assets to some encryption method.”

…to which he retorted that I should have included these points in the first place.

In my defense, there was a cap based on word count, and I made cuts where I felt appropriate.  I guess I was wrong.

I think it’s absolutely fucking ridiculous that in all of my requests for financial help over the years, I always get nailed for making too much money.

What do institutions want from me?  Oh, right, I know: be destitute so that I can be eligible for the scraps that they offer me, as if they are helping me by forcing me into a corner.  As if I were lying about the financial costs, and demands on my time as a caregiver for elderly parents with no other family and no other resources.  I work and make an honest living, but apparently, I’d only be eligible for help if I didn’t, and were on welfare.

Well, fuck you, system.  Fuck you.  Just say what you really want to say: I’d rather you go into debt than provide help to you in developing a foundation in life from which you could actually potentially build a future.  This is because I let poor people just barely stay afloat.  If we helped people who worked for a living, well, they might actually get somewhere.

These are the real messages you’re sending with the rules that you have set up.

NB:  My opinion on going into debt for education is that you shouldn’t.

The problem of induction was introduced by David Hume (1711-1776) and started with the question of whether or not induction is justified. This is a genuine concern since predictions about the unobserved/future that are derived from experience are made through inductive inference, and are not deductively closed arguments (i.e. not a priori knowledge).

To illustrate the problem, let us begin with the following example:

In my experience, all F’s are G’s, and no cases of F’s have been found to not be G’s.

I arrive at the general statement that “All F’s are G’s” through inductive reasoning.

Is this generalization “justified”?  It is immediately clear that the generalization does not necessarily follow from the premise, since it is not arrived at deductively (that is, it is not entailed).  So it appears that we take a leap from premise to generalization when we reason inductively.

Upon careful examination of the above example, the generalization can be justified by the apparent “Uniformity of Nature”, as discussed by Bertrand Russell in The Problems of Philosophy, Chapter VI.  ”The belief in the uniformity of nature is the belief that everything that has happened or will happen is an instance of some general law to which there are no exceptions.”  Herein lays the problem: uniformity of nature is a premise that can only be arrived at inductively, so it cannot be used to justify inductive reasoning.  It is a circular argument.

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P. F. Srawson attacks this problem from a linguistic standpoint, claiming that the question of whether or not induction is justified is nonsensical.  He says that it is “the absurd wish that induction should be shown to be some sort of deduction.”  His argument is illustrated as follows:

To be rational is to use induction and deduction.

Deductive and inductive reasoning are mutually exclusive.

The word “deductive” describes closed arguments that lead to a priori knowledge.

Deductive propositions are therefore either valid, or invalid on the basis of being either justified or not justified.

The word “inductive” describes the reasoning that leads to degrees of belief that are supported by experience.

So, questions such as “Is there reason in believing in deductive arguments?” and “Are inductive arguments justified?” have no meaning for Strawson.

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Karl Popper attempts to show that “the belief that we use induction is simply a mistake. [...] The whole apparatus of induction becomes unnecessary once we admit the [...] conjectural character of human knowledge.”  He discards induction with his notions of the following:

1. Although we cannot employ induction to acquire a necessary truth, we can necessarily conclude the falsity of a generalization with falsifying evidence, and this is purely deductive.

2. Laws arrived at inductively were based on “unconscious, inborn expectations” or “scanty material, i.e. the few observed instances upon which the law may be based.”

He proposes that conjectures (hypotheses) are arrived at arbitrarily, either through myths, or inborn expectations, and that testing (trying to find refutations) is how one arrived at conjectures with (degrees of) corroboration, as opposed to inductive inferences with (degrees of) probability.  So, he discards induction, but only to appeal to it in different terms.

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Neither attack of the problem is satisfying.  No one said it better than Russell when he said that “we must either accept the inductive principle on the ground of its intrinsic evidence, or forgo all justification of our expectations about the future.”  And all the while philosophers who attack this problem tragically become Kierkegaard’s Knights of Infinite Resignation: afraid and too calculating to take the leap that they miss the point.

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Postscript:

This was my first assignment after returning to school after an extensive hiatus . This means it was written around.  It’s just a short little ditty, and I stumbled upon it while going through my old class notes.  I was awarded a 90%, and a comment regarding my criticism of criticisms on the problem of induction.  I don’t think the T.A. who marked it appreciated that I called people in his profession Kierkegaard’s Knights of Infinite Resignation. I was hoping he’d get a kick out of it.

Colyvan described the Indispensability argument as “the best argument for Platonism.”  Its basic structure, as per Colyvan:

    We ought to have ontological commitment to all the entities that are indispensable to our best scientific theories.  (Confirmational Holism)
    We ought to have ontological commitment to only the entities that are indispensable to our best scientific theories.  (Naturalism)

———————————————————————————————————–

    (Premise1) We ought to have ontological commitment to all and only the entities that are indispensable to our best scientific theories.
    (Premise2) Mathematical entities are indispensable to our best scientific theories.
    ———————————————————————————————————–
    (Conclusion) We ought to have ontological commitment to mathematical entities.

Field objects to Premise2.  His argument has two parts.  The first is that mathematical theories don’t have to be true to be useful, they need only be conservative. Conservatism refers to this result: if A is a consequence of T (a scientific theory) + M (Mathematics), then A would be a consequence of T alone (Brown, pg58).  Mathematics is a useful tool, but it is not indispensable.  The second part of Field’s program is to demonstrate that our best scientific theories can be suitably nominalised. By nominalising a portion of Newtonian gravitational theory, he attempts to show that there is no need to assert the existence of mathematical entities in a scientific theory. This is not trivial example – the hope is this example can represent the greater case of all scientific theories.

Field’s objection does strike the Indispensability argument with a good blow.  However, as noted by Brown (pg. 59), “the notion of logical consequence that is needed is that of second-order logic” which is not recursively axiomatizable.  This means that the notion of consequence is not “nominalistically acceptable” since it involves being true in all models.  Also, if Maddy’s program pans out, then Field’s objection is irrelevant.

Realism vs. Anti-Realism and Models of The Atom

Introduction

Did physicists believe in the reality of their atomic models? To answer this question, one has to look first at what it means to say that one ‘believes in the reality’ of something, and then also understand what is meant by a scientific ‘model’.  (From this point forward, I will refer scientific realism and scientific anti-realism as “realism” and “anti-realism”, respectively.)  “Discerning the aims of physical theory has been an important goal since the Greeks, with realist rather than positivist or instrumentalist views dominating at one time or another.”[i] “For nearly all practicing scientists—not all, to be sure—realism is an unequivocal commitment, rarely reflected upon very deeply.  Science, according to this view, is not merely another cultural activity, not simply fashion or metaphor, not simply an alternative way of viewing the world.  The success of science, its efficacy, its law-giving character—indeed the “progress” of science—clearly distinguishes it from other, no less important, areas of human inquiry.”[ii] The realist makes two claims:

  1. “Scientists ought to seek to formulate true theories that depict the structure of the universe…[and oppose] instrumentalists…who sought to restrict science to the “saving of appearances”.[iii]

and

  1. “The record of progress indicates that the universe has a structure (largely) independent of human theorizing and that our theories have provided an increasingly more accurate picture of that structure.”[iv]

There are other forms of realism, such as Entity Realism as propounded by Ian Hacking, and Structural Realism by John Worral, that are weaker versions of the realist position.  They don’t require that all of scientific practice aims for and attains truth and knowledge of reality in itself, and that the development is science is progressive.  They just pick out parts that could be so – such as putative entities, and structures.  Anti-realism can take many forms, but at the very least an anti-realist “seek(s) to uncouple the notions of predictive success and truth.”[v] Instrumentalism is a form of anti-realism that says that “scientific theories are calculating devices that facilitate the organization and prediction of statements about observations. [...] Theories are merely “useful” or “not useful”.”[vi] Bas Van Fraassen is a Constructive Empiricist which is a form of Instrumentalism, and he maintains that the goal of science is to “formulate empirically adequate theories… [not to] … establish the truth of claims about theoretical entities.”[vii] Models can be both realist and anti-realist.  They “have two main functions in physics: they may be proposed either as putatively true representations of the physical characteristics of the objects treated by some theory, or as purely imaginary devices, heuristic fictions (a formal model).”[viii] In either case, whether proposed as putatively true or as a heuristic device, models are suggestive.

In the case of the atom, there was a full range of ontologies that were adopted by practicing physicists.  Developments in the nineteenth century culminated in the development of electromagnetic field theory with Maxwell, and statistical mechanics with Boltzmann.  These two physicists were realists – they believed that their models of the atoms mapped onto reality.  They did have opposition among their contemporaries, such as anti-realist Ernst Mach.  Mach was an anti-realist about unobservable entities.  He had a positivistic approach to the science, and since physical theory during their time did not require the existence of the atom, he did not adopt belief in it.  As regards twentieth century physics, following the Quantum Revolution, there were two main interpretations of Quantum Mechanics that provided models for interpretation of the mathematics, and consequently the atom.  However, following the Solvay Congress in October 1927, the Copenhagen Interpretation as given by Bohr and Heisenberg came to dominate as the most accepted one.[ix] In my paper, I will argue that the question of the ontological status of the atom changed from “does it exist at all (Maxwell/Boltzmann)?”  to a question of “given the atom, what is its nature?” (Bohr/Heisenberg).  Particularly, that there was a shift after 1905 from realist to antirealist attitudes towards the ontological status of the atom.

Nineteenth Century Atomism

Positivist thought “began to be felt at the end of the nineteenth century, promoted by Comte, the Vienna Circle, and the scientist-philosopher-historians such as Pierre Duhem and Ernst Mach.”[x] And this made sense because “before atomic theory became firmly established, and when physics could study only macroscopic phenomena, mechanical models and speculative hypotheses about underlying structure could be counterproductive.  A theory might fail because of such a model, while a macroscopic model only had to describe or reproduce the phenomena.”[xi] “In a loose sense, the distinction between dynamists and mechanists was one between positivists and realists, even though the ideas are not equivalent.  A positivist essentially sees the aim of physical theory as economically summarizing empirical results: as the Greeks saw it, “saving the appearances.”  No mechanical hypotheses are introduced that are not justified by what is observable.  The realist, of course sees the entities introduced to explain the experimental results as objectively real.  The mechanist, in trying to explain the properties of matter on the basis of the nature of its smallest parts, often has recourse to entities that are not accessible to observation.  In some cases entities may ultimately be observed and become part of the empirical world; in others they may disappear from the literature or survive only as heuristic elements.  In some cases, of course, the entities are not intended to be real and serve only as analogy, as an aid in reasoning.  This is sometimes the case in Maxwell’s use of models, which included elements that he never claims to exist fully.  Yet this is no doubt of Maxwell’s commitment to the reality of the molecular vortices on which much of theory of electromagnetism was based.”[xii]

James Clerk Maxwell and Ludwig Boltzmann were realists.  They “did believe in the [realty of their atomic] model(s), particularly in [...] the molecular vortical model.”[xiii] Their collaboration led to the development of a model of the atom where “the particularly simple properties of a molecular model, according to which the molecules are point masses (thus not hard sphere) which interact with a repulsive force inversely proportional to the fifth power of their distance.”[xiv] Their model was visualizable, and was explainable within the current paradigm.  With a visualizable model, they were able to use of analogy to guide their investigations.  “The role of analogy in nineteenth-century physics [...]was used deliberately and self-consciously by some of the most important scientific figures of the time, especially [...] Maxwell.  Indeed, Maxwell not only used a method of physical analogy with great success but also speculated extensively about it, especially the question of whether analogies in the natural world or the human mind. [...] Maxwell employed mechanical models to whose reality he was committed in differing degrees at different times.”[xv] As for Boltzmann’s commitment to his atomic model, in a letter, he wrote: “The realist compares the assertion that he could never imagine how the mental could be represented by the material let alone by the interaction of atoms with the opinion of an uneducated person who says that the Sun could not be 93 million miles from the Earth, since he cannot imagine it.  Just as the ideology is a world picture only for some but not for humanity as a whole, so I think that if we include animals and even the Universe the realist mode of expression is more appropriate.”[xvi]

Twentieth Century

Philosophical positivism was evident in the late nineteenth century practice of physics in the opinions of such physicist/philosophers such as Ernst Mach, and his rejection of atomism.  However, it was the operationalist character of quantum ontology of Neils Bohr’s in the twentieth century that also reflects philosophical positivism.[xvii] An operationalist says that “it is the operations by which values are assigned that give empirical significance to a scientific concept.”[xviii] “Though they were verbally opposed to several theses of the positivism of the philosophers [in the Vienna Circle], the physicists of the Copenhagen School, for their part, built up a quantum mechanics in which certain lines of reasoning when followed closely suggested … rather similar views.”[xix] And although it was not the only interpretation of Quantum Mechanics, following the Solvay Congress of 1927, the Copenhagen Interpretation of Quantum Mechanics was the dominant one.

As stated, Bohr was a positivist.  He took an instrumentalist’s viewed toward his atomic model.  He was “extremely cautious.  He believe(d) that the models of atomic structure have some realistic significance, but he is acutely conscious of the negative analogy of the models; indeed he doubts whether a complete, realistic model of atomic processes is obtainable. [...] His own anti-realism was inspired by his commitment to Machian positivism. [...] Models help us to construct theories which enable us to explain and predict the course of our sensory experience.  Highly successful models may owe their success to the fact that they faithfully represent at least some aspects of the real entites which lie beneath the appearances.  … We ought not to put too much  faith in the realistic performance of models.”[xx] In a letter to Hoffding, he very tellingly wrote:

“The question of the role of analogy in scientific investigations which you stressed is undoubtedly an essential feature of every study in the natural sciences, even if it does not always stand out.  It is often quite possible to make use of a picture of a geometrical or arithmetical sort which covers the problem in question in such a clear way that the considerations almost acquire a purely logical character.  In general, and particularly in some new fields of research, one must however constantly keep in mind the obvious or possible inadequacy of the picture, and , so long as the analogies make a strong showing, be content if the usefulness or rather fruitfulness in the area they are used is beyond doubt.  Such a state of affairs holds not least from the standpoint of the present atomic theory.  Here we are in the peculiar situation that we have gained some information about the structure of the atom which may safely be considered just as certain as any of the facts of natural science.  On the other hand we meet with difficulties of such a profound nature that we cannot see any way of solving them; in my personal opinion these difficulties are of such a kind that they scarcely allow us any hope of carrying through in the atomic realm a description in time and space of the kind that matches our ordinary sense impressions.  In these circumstances one must naturally bear in mind that one is operating with analogies, and the point, that the areas of use of these analogies in the individual case are restricted, is of decisive importance for progress.”[xxi]

He wrote this during the decline of his original atomic model, which was already being found to be flawed.

Bohr and Heisenberg’s model Copenhagen Interpretation of Quantum Mechanics depicted the atom as no longer visualizable.  In this interpretation, there is no quantum world that exists independently of our observation.  The observer and observed are inseparable, and that to make a measurement is to define the operation performed in making that measurement.  All that is knowable is what you observe, and what you observe is affected by your action of making the observation.  In adopting instrumentalist views toward their atomic model, they acknowledged the failures of using a model for visualization.  This sort of limitation of a model is analogous to the failures of using a tesseract or hypercube as a 3-dimensional representation of a 4-dimensional object.  Where some useful inferences can be drawn, there may also be ones that fail simply because the model failed.

Concluding Remarks

Nineteenth century and twentieth century physics had entirely different climates.    We’ve looked at nineteenth century realists, and twentieth century instrumentalists, but “there is no single scientific method that [was] applicable in all fields and at all times or to both theorists and experimentalists”[xxii] There were nineteenth century anti-atomists such as Pierre Duhem and Ernst Mach and twentieth century realists such as Einstein.  But there was definite change in attitude with the Quantum Revolution.  Older texts wrote of the way that “independent reality refuses to tell us what it is – or what it is like – it at least condescends to let us know, to some extent, what it is not.  It does not conform to the classical schemes of mechanics, of atomistic materialism, or of objectivist realism – in short, to any variant of ‘near realism’.[xxiii] And textbooks on twentieth century physics stress that “most [scientists] assign a more modest goal to physics, and to knowledge in general.  Science, they say, (and ordinary knowledge as well) is indissolubly linked with human experience.  Once and for all it must therefore give up the unattainable goal of describing whatever some thinkers may mean when they speak of ‘reality in itself’ or ‘reality as it really is’.  The task of science can only be a description of the phenomena, that is, of things, events and so on, as they are organised by human collective experience.”[xxiv]

It was the kinetic theory of gases that changed the ontological status of the physical atom from speculation to reality, and “the understanding of atomic and molecular spectra achieved by the 1860’s and 1870’s, which made possible the use of spectroscopy in chemical analysis, went far toward bridging the gap between the two manifestations of the microscopic structure of matter.  The final resolution came only with a detailed theory of atomic structure, which had to wait for…the quantum revolution.”[xxv]

“But for the most part, the nineteenth century ended the [...] debates about the microscopic structure of matter and provided convincing proof of the reality of the atoms. [...] Models of the internal structure of the atom were being seriously proposed. [...] Some would say that the final blow to the opponents of atomism was Einstein’s 1905 paper on Brownian motion, which showed that it was due to the motion of molecules.”[xxvi] “It is not too much to say that the great revolution in twentieth-century physics—the quantum theory—owes its birth to atomism, not merely in the strict historical sense but because the nineteenth-century success of the corpuscular theory prepared the way for the discontinuities and quantization that lie at the heart of quantum theory.”[xxvii] In this way, we can say that the realist attitudes of Maxwell and Boltzmann paved the way for the instrumentalism of Bohr and Heisenberg that were to come.

Bibliography

Bowler, Peter J. 2005. Making modern science : A historical survey, ed. Iwan Rhys Morus. Chicago: University of Chicago Press.

Buchwald, J. Z. A Brief History of Electric and Magnetic Science (unpublished)

Cassidy, David C. 2008. Beyond uncertainty : Heisenberg, quantum physics, and the bomb. New York: Bellevue Literary Press.

Cercignani, Carlo. 1998. Ludwig boltzmann : The man who trusted atoms. New York: Oxford University Press.

Espagnat, Bernard d. 1989. Reality and the physicist : Knowledge, duration, and the quantum world. New York: Cambridge University Press.

Great experiments in physics : Firsthand accounts from galileo to einstein(1987). In Shamos M. H. (Ed.), . New York: Dover Publications.

Kevles, D. J. (1978). The physicists : The history of a scientific community in modern america. New York: Knopf.

Kragh, H. (1999). Quantum generations : A history of physics in the twentieth century. Princeton, N.J.: Princeton University Press.

Losee, John. 1980. A historical introduction to the philosophy of science. 2d ed. — ed. London: Oxford University Press.

Murdoch, Dugald. 1987. Niels bohr’s philosophy of physics. New York: Cambridge University Press.

Purrington, R. D. (1997). Physics in the nineteenth century. New Brunswick, N.J.: Rutgers University Press.

Wilson, David. 1983. Rutherford, simple genius. London: Hodder and Stoughton.


[i] Purrington, Pg. 19

[ii] Purrington, Pg xi

[iii] Losee. Pg. 253

[iv] Losee. Pg. 253

[v] Losee, Pg. 254

[vi] Losee, Pg. 257

[vii] Losee, Pg. 257

[viii] Murdock, Pg. 74

[ix] Kragh, pg.212-215

[x]Purrington, g. 7

[xi] Purrington, Pg. 21

[xii] Purrington, Pg. 22

[xiii] Purrington, Pg. 67

[xiv] Cercignani, Pg. 199

[xv] Purrington, Pg. 28

[xvi] Cercignani, pg 174

[xvii]Purrington, g. 7

[xviii] Losee, Pg. 160

[xix] D’Espagnat, Pg. 200

[xx] Murdoch, pg 76-77

[xxi] Murdoch, pg 76

[xxii] Purrington, Pg xii

[xxiii] D’Espagnat, Pg. 208

[xxiv] D’Espagnat, Pg. 232

[xxv] Purrington, pg. 131

[xxvi] Purrington, pg. 131

[xxvii] Purrington, Pg. 131