Numerical Analisys and Epistemology of InformationPhilosophical Aspects of Computer Science

Dr Marco Benini

marco.benini@uninsubria.it

Dipartimento di Scienza e Alta TecnologiaUniversità degli Studi dell’Insubria

March 6th, 2015

Introduction

Computational numbers 6= mathematical numbers

■ mathematical numbers carry a structure with them which is radicallydifferent than the one of computational numbers

■ numerical algorithms operate on computational numbers, although theyare justified within a mathematical framework

■ numerical algorithms solve real-world problems whose observables aremeasured by numbers in a mathematical model

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Introduction

LoA

real world ⇔ observables

↓ ↓mathematical model ⇔ numbers, algorithm

↓ ↓computational model ⇔ numbers, program

LoO

In L. Floridi’s terms, we have Levels of Abstraction (LoAs), and Levels ofObservation (LoO). These levels come in pairs, and the abstraction processinduces a pair of maps.The abstraction process assigns a meaning to a LoA/LoO in terms of themaps from the more concrete LoA/LoO. In practise, a computation in themore abstract level simulates a piece of the real world and its output isinterpreted in the real world reversing the abstraction map(s) in the LoO.

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Computational numbers

■ at least two ways to interpret them: as numbers and as strings of bits■ finite objects to fit into the computer memory■ in most cases bounded objects, so, e.g., there is a maximum and aminimum number (!)

■ efficient manipulation by the computer hardware■ within well-specified bounds, they behave as (approximations of)mathematical numbers

RemarkComputational numbers are formal mathematical objects, but of a differentsort which is loosely related to proper mathematical numbers.

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Structure and paradoxes

ExampleAn apparent paradox: addition is not associative, even if it is

f (x ,y ,z)= x + (y +z) g(x ,y ,z)= (x +y)+z

In mathematics, f = g when x , y , z are numbers (N, Z, R, C).But, if computational numbers are formed by two decimal digits:

f (97,4,−5) = 97+ (4−5)= 97−1 = 96g(97,4,−5)= (97+4)−5= ??−5= ??

Numerical stability and related phenomena are well-known to numericalanalysts: these aspects must be considered in the implementation ofalgorithms.Moving from the mathematical LoA to the computational LoA means toabstract away part of the structure of numbers: the correctness ofalgorithms is preserved as far as the structure justifying them is retained.

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Correctness

ProblemGiven a pair (a,b) in R such that f (a)< 0, f (b)> 0, with f continuous andmonotone, and assuming that there is a unique root R of f in the interval[a,b], calculate it.

-q q qa bRr

f

qNewton’s method:let r = (a+b)/2if f (r)> 0 iterate on (a,r)if f (r)< 0 iterate on (r ,b)if f (r)= 0 stop

It is possible to show that r approximates the root of f better and better, upto any degree of precision. Thus, Newton’s method is correct.

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Correctness

Consider

-q q qa bR

r

f

qSince f (r)< 0 but it is very close tozero, its representation may be 0 oreven a small positive number. In bothcases, the iteration of the method willfail to refine the approximation of R,as |R − r | will not decrease.

So Newton’s method is computationally wrong!■ we may think, as usual in numerical analysis, that the calculated f (r)contains an error causing the method’s failure;

■ or, we may think that r and f (r) are points with some degree ofuncertainty: when the uncertainty of f (r) is big enough, it prevents thecase choice in the method, causing its failure.

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Measures

The relation between the real world and the mathematical LoA shows similardifficulties: an act of measuring is required to transform an observable into anumber. And, in most cases, the uncertainty of the measure is poorlyconsidered.Many examples from physics are immediate to show, leading to paradoxicalresults: the truth is that there is a long withstanding tradition in physics andengineering that analyses these problems and provide solutions.But, nevertheless, computer science uses many other mathematical models,where the problem is, at least, underestimated.

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Paradoxes

ExampleWeb banking applications are very popular, nowadays. If a user has a bankaccount, a savings account, and some investments like bonds or stockoptions, the application usually shows the total capital. Strictly speaking,this number is wrong.In fact, to use money, the user has to transfer it, by some means, tosomeone else: the nature of money is to allow an exchange. But bonds andstock options are not money: they must be sold (converted) before they canbe used to buy something. And their value (the result of the conversion)depends on the state of the market in the instant they are sold. Moreover,the selling act modifies their value.So, the capital, as shown by the web banking application, contains anamount of uncertainty, given by an estimate of the value of the investmentsas if they were sold at the moment the grand total is calculated.

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Paradoxes

ExampleTests are a popular form of evaluation: their appeal is that the check can beautomated. Suppose a test form contains one hundred of questions, andeach question has four possible answers, only one being correct. Let usstipulate that the score of the test is the sum of the correct answers.If a student has not prepared the examination, and thus gives randomanswers, she will get a score close to 25. Now, a not-too-bad student mayreasonably get a score around 90. Nevertheless, this student is virtuallyindistinguishable from the cheater who has studied very well just 85% of theprogram, and who randomly guesses the questions she doesn’t know about.

As before, the problem lies in the uncertainty of the measure: giving thecorrect answer to a question does not imply the student knows it!Of course, correctly answering to many questions provide an (uncertain)evidence of knowledge.

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Representation and abstraction

Although very different one from the others, all the examples show that amathematical or computational number represent some more concrete entityin the lower LoA, and the abstraction process throws away an intrinsicuncertainty, either present in the entity, or in the process leading to therepresentation.It is exactly the inability to recover from the numbers the uncertainty(whatever form it takes) that allows to construct apparently bizarre,nonsensical, wrong, or paradoxical examples.But the positive news is that there are at least two ways to overcome thesesituations, arising from the philosophy of information and contemporarymathematics.

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Levels of explanation

A Level of Explanation (LoE) is a presentation of a LoA, LoO pair tailoredtowards a specific purpose.In the cases under inspection, the LoE of interest is the one which explainsthe numbers as representations of quantities in the lower LoA.For example, a LoE may explain how a real number becomes a floating pointnumber, showing that the mapping is not injective and, thus, identifying afloating point number as the counter-image of the representation map. Inthis way, it is clear that a computational number is, indeed, an interval ofreals, thus obeying to a different algebra.In essence, this is the philosophical description of the approach adopted inphysics, and it explains why it works.

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Numbers as byproducts

An alternative way to cope with the difficulties illustrated in this talk is toabandon number as primitive concepts: instead, we may use point-freemathematics to model the problem and symbolic computation to implementthe mathematical algorithm arising from the model.Although this kind of mathematics is more abstract, the numbers, which area derived notion in this setting, carry with them their construction process,which may be used to model uncertainty. Since the construction process ofnumbers can be hidden, the resulting algorithms and programs are verysimilar to the ones obtained in more traditional frameworks, but,nevertheless, they are enriched with a notion of uncertainty which can beused at need.Thus, uncertainty is no more uncertain, solving the difficulties we illustratedso far, at least when the technique can be applied.

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Conclusion

The moral of this talk is

Uncertainty is what make numerical computations to fail

Uncertainty is a piece of knowledge that has been forgotten in theabstraction process from the real world, where the problem to solve lies, tothe computational world, where the solution is calculated.The proposal in this talk is not to forget but to hide, so to decreaseuncertainty, either by using the epistemological structure (LoE) behind thedescription of the process leading to the solving program, or by adopting aricher and more sophisticated theory and using it to develop themathematical model.No solution is better, in principle, and there are successful examples of both:at the present stage, it seems largely dependent on the nature of theproblem (how?) which solution works best (why?).

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Questions?

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