1. Numerical Analysis and Epistemology of Information
Philosophical Aspects of Computer Science
Dr Marco Benini
marco.benini@uninsubria.it
Dipartimento di Scienza e Alta Tecnologia
Università degli Studi dell’Insubria
March 6th, 2015

2. Introduction
Computational numbers = mathematical numbers
mathematical numbers carry a structure with them which is radically
different than the one of computational numbers
numerical algorithms operate on computational numbers, although they
are justified within a mathematical framework
numerical algorithms solve real-world problems whose observables are
measured by numbers in a mathematical model
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3. 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 of
Observation (LoO). These levels come in pairs, and the abstraction process
induces a pair of maps.
The abstraction process assigns a meaning to a LoA/LoO in terms of the
maps from the more concrete LoA/LoO. In practise, a computation in the
more abstract level simulates a piece of the real world and its output is
interpreted in the real world reversing the abstraction map(s) in the LoO.
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4. 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 a
minimum number (!)
efficient manipulation by the computer hardware
within well-specified bounds, they behave as (approximations of)
mathematical numbers
Remark
Computational numbers are formal mathematical objects, but of a different
sort which is loosely related to proper mathematical numbers.
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5. Structure and paradoxes
Example
An 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 = 96
g(97,4,−5) = (97+4)−5 = ??−5 = ??
Numerical stability and related phenomena are well-known to numerical
analysts: these aspects must be considered in the implementation of
algorithms.
Moving from the mathematical LoA to the computational LoA means to
abstract away part of the structure of numbers: the correctness of
algorithms is preserved as far as the structure justifying them is retained.
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6. Correctness
Problem
Given a pair (a,b) in R such that f (a) < 0, f (b) > 0, with f continuous and
monotone, and assuming that there is a unique root R of f in the interval
[a,b], calculate it.
-q q qa bRr
f
q
Newton’s method:
let r = (a+b)/2
if 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, up
to any degree of precision. Thus, Newton’s method is correct.
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7. Correctness
Consider
-q q qa b
R
r
f
q
Since f (r) < 0 but it is very close to
zero, its representation may be 0 or
even a small positive number. In both
cases, the iteration of the method will
fail 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 of
uncertainty: when the uncertainty of f (r) is big enough, it prevents the
case choice in the method, causing its failure.
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8. Measures
The relation between the real world and the mathematical LoA shows similar
difficulties: an act of measuring is required to transform an observable into a
number. And, in most cases, the uncertainty of the measure is poorly
considered.
Many examples from physics are immediate to show, leading to paradoxical
results: the truth is that there is a long withstanding tradition in physics and
engineering 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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9. Paradoxes
Example
Web banking applications are very popular, nowadays. If a user has a bank
account, a savings account, and some investments like bonds or stock
options, 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, to
someone else: the nature of money is to allow an exchange. But bonds and
stock options are not money: they must be sold (converted) before they can
be 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 an
amount of uncertainty, given by an estimate of the value of the investments
as if they were sold at the moment the grand total is calculated.
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10. Paradoxes
Example
Tests are a popular form of evaluation: their appeal is that the check can be
automated. Suppose a test form contains one hundred of questions, and
each question has four possible answers, only one being correct. Let us
stipulate that the score of the test is the sum of the correct answers.
If a student has not prepared the examination, and thus gives random
answers, she will get a score close to 25. Now, a not-too-bad student may
reasonably get a score around 90. Nevertheless, this student is virtually
indistinguishable from the cheater who has studied very well just 85% of the
program, and who randomly guesses the questions she doesn’t know about.
As before, the problem lies in the uncertainty of the measure: giving the
correct 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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11. Representation and abstraction
Although very different one from the others, all the examples show that a
mathematical or computational number represent some more concrete entity
in the lower LoA, and the abstraction process throws away an intrinsic
uncertainty, either present in the entity, or in the process leading to the
representation.
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 these
situations, arising from the philosophy of information and contemporary
mathematics.
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12. Levels of explanation
A Level of Explanation (LoE) is a presentation of a LoA, LoO pair tailored
towards a specific purpose.
In the cases under inspection, the LoE of interest is the one which explains
the numbers as representations of quantities in the lower LoA.
For example, a LoE may explain how a real number becomes a floating point
number, showing that the mapping is not injective and, thus, identifying a
floating point number as the counter-image of the representation map. In
this way, it is clear that a computational number is, indeed, an interval of
reals, thus obeying to a different algebra.
In essence, this is the philosophical description of the approach adopted in
physics, and it explains why it works.
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13. Numbers as byproducts
An alternative way to cope with the difficulties illustrated in this talk is to
abandon number as primitive concepts: instead, we may use point-free
mathematics to model the problem and symbolic computation to implement
the mathematical algorithm arising from the model.
Although this kind of mathematics is more abstract, the numbers, which are
a derived notion in this setting, carry with them their construction process,
which may be used to model uncertainty. Since the construction process of
numbers can be hidden, the resulting algorithms and programs are very
similar to the ones obtained in more traditional frameworks, but,
nevertheless, they are enriched with a notion of uncertainty which can be
used at need.
Thus, uncertainty is no more uncertain, solving the difficulties we illustrated
so far, at least when the technique can be applied.
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14. 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 the
abstraction process from the real world, where the problem to solve lies, to
the computational world, where the solution is calculated.
The proposal in this talk is not to forget but to hide, so to decrease
uncertainty, either by using the epistemological structure (LoE) behind the
description of the process leading to the solving program, or by adopting a
richer and more sophisticated theory and using it to develop the
mathematical 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 the
problem (how?) which solution works best (why?).
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15. Questions?
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