Towards a critique of digital reason

C. M. Sperberg-McQueen, Black Mesa Technologies LLC

23 July 2019

http://blackmesatech.com/2019/07/Leipzig/

Overview

What is the question?

The question

Kant's starting point

Wir dürfen aber die Möglichkeit solcher Sätze hier nicht zuerst suchen, d.i. fragen, ob sie möglich sein. Denn es sind deren gnug, und zwar mit unstreitiger Gewißheit wirklich gegeben, und ... so werden wir davon anfangen: daß dergleichen ... Vernunfterkenntnis wirklich sei;
We do not first need to ask after the possibility of such propositions [sc. synthetic a priori propositions], i.e. ask whether they are possible. There are plenty of them, with indisputable certainty, and ... we will thus begin with the observation that such ... cognitive achievements of reason are real;

Kant's starting point (2)

aber alsdenn müssen wir den Grund dieser Möglichkeit dennoch untersuchen, und fragen, wie diese Erkenntnis möglich sei, damit wir aus den Prinzipien ihrer Möglichkeit die Bedingungen ihres Gebrauchs, den Umfang und die Grenzen desselben zu bestimmen in Stand gesetzt werden.
but then we must immediately investigate the reason for this possibiity, and ask how this cognition is possible, in order that on the basis of the principles governing its possibility we can derive the conditions of its use, its amplitude and its boundaries.

Our goals

Warning

I am not Kant.
I am not an expert on Kant.
I am not a philosopher.

A priori forms, categories

What is digital reason?

Two premises

As to a priori forms of perception:
  • Digital reason can take into consideration only what we can represent digitally.
As to categories of thought:
  • Digital reason can only reach results accessible to combinations of its basic operations.
    (Or: it can only arrive at conclusions accessible to its underlying logic.)

What can we represent digitally?

Space and time?

Space and time are obviously relevant for computation.
Do they limit digital and human reason in the same way?
No — Kant had something different in mind.

Image illustrating 3-D Euclidean space

Space and times are forms of perception: whatever we perceive, we perceive in space and time.

Space as a form of perception

If we don't perceive it in space and time,
we don't perceive it.
Wittgenstein puts it clearly:

Jedes Ding ist ... in einem Raume .... Diesen Raum kann ich mir leer denken, nicht aber das Ding ohne den Raum.

Every object is ... in a space .... I can imagine the space being empty, but I cannot imagine the thing without the space.

Is space an a priori form of perception?

Image illustrating idea of curved space

Prerequisites (1): digital representations

To reason digitally about anything (a value in a domain),
we must represent that thing digitally.
Requirements:
  • mechanical (physical) representations, so machine can operate on them.
  • creatable: Both humans and machines must be able to create representations.
  • stable: Representations must be stable enough to persist until needed again.
  • finite: In practice, representations of values must be finite
    • in size,
    • in number.
  • expressive (enough): Representations must allow us to distinguish the values we need to distinguish.
This is where we may expect inescapable limitations to arise.

The digital/analog distinction

We've been talking about mechanical representations.
But the topic specifies digital reason.
What does it mean to say “Digital reason can operate only on propositions digitally represented”?
  • digital is not (just) a hip way to say “electronic” or “computer-based”.
  • The antonym of digital is not in-real-life but analog.

Von Neumann on the digital/analog distinction

John von Neumann describes the digital/analog distinction this way:
Existing computing machines fall into two broad classes: ‘analog’ and ‘digital’. This subdivision arises according to the way in which the numbers, on which the machine operates, are represented in it.
In an analog machine each number is represented by a suitable physical quantity, whose values, measured in some pre-assigned unit, is equal to the number in question. This quantity may be the angle by which a certain disk has rotated, or the strength of a certain current, or the amount of a certain (relative) voltage, etc.
...
In a decimal digital machine each number is represented in the same way as in conventional writing or printing, i.e. as a sequence of decimal digits. Each decimal digit, in turn, is represented by a series of ‘markers’.

Analog representations

In an analog representation:

Image showing a dial with a needle to register pressure in a pump

  • In principle, infinite number of distinct possible representations, representing infinite number of distinct values.
  • Between any two representations (points on the dial), there is a third with an intermediate value.
  • Continuum on the dial represents continuum of real numbers.

Digital representation

In a digital representation, by contrast:
  • Finite number of distinct possible representations
  • representing finite subset of infinite number of distinct values.
  • Meaningless to speak of any representation lying “between” two others.
  • Is there a representation between
    123
    124
    125
    ?
  • How about
    ៥៦៧
    ៥៦៩
    ៥៦៨
    ? Can anyone read Khmer numerals?

Betweenness

Consider three numerals (digital representations of numbers):
  • 1000 0000 = -256
  • 0001 0000 = 16
  • 0000 0001 = 1
  • The second representation is “between” the other two (for a natural interpretation of “between”).
  • But the second value is not between the other two values.
Digital representations never guarantee that between any two representations lies a third.
Why? Because the concept of ‘between-ness’ does not apply.
Digital representations never guarantee that between any two representatble values lies a third.
Digital representations can approximate values from continuous sets,
but they do not mirror the all properties of the values or of the set.

Approximation

Approximation is inescapable in digital representations.

Approximation (2)

If we cannot be exact, why bother?
  • Most fields need only a few digits of precision.
  • Analog representations also have limited precision.
  • Adding precision is cheaper for digital machines than for analog machines.
  • For finite sets of values, exact representation is possible (for digital, but not for analog, representations).
  • Language, writing, and text are essentially digital.
    • Sounds are continuous; phonemes are digital.
    • Script is continuous; graphemes are digital.
    • Continuous variation in sound is quantized when we recognize phonemes.
    • Continuous variation in shapes on page is quantized when we recognize graphemes.

Digital representation and understanding

Understanding speech and writing
consists
in converting an analog signal (sounds, marks on paper)
into a digital signal (phonemes, graphemes, words, sentences).

Digital representation and size

An analog representation of a page (67 kB):

Picture of one page of Athanasius Kircher's book Ars magna sciendi

Digital representation (transcription): 2 kB

Digital vs analog

Digital representations (e.g. transcriptions) are selective:
they retain meaningful information and discard non-meaningful information.
Analog representations (e.g. scans)
require less knowledge, discard less information.
Neither replaces the other.
But in general, the more certain we are that we understand the information, the happier we will be with digital representations.

Analog representations are not complete

Analog representations discard less information.
But that does not mean they are complete. They are not and cannot be.

What can we not represent digitally?

What kinds of things cause trouble for digital representation?

Image showing the color plane

Any continuous space (e.g. color space).

What can we not represent digitally? (2)

What kinds of things cause trouble for digital representation?

Image of puffy clouds across the sky

Any space with unordered, non-discrete values. (E.g. clouds).

Prerequisites (2): operations

Static representations of values are not enough.
We need to be able to process them. Representations must be:
  • processable: We need mechanical operations on values which correspond to operations in the domain.
  • correct: Operations should produce correct results.

Operations in the application domain

In the application domain, operations on values produce new values.

Image showing two values in a domain feeding into an operation and producing a third value

Digital representation of operations

Digital processing mirrors operations at physical level.

Image showing physical representations of the two input values, the operation, and the third result value, all linked by the 'Represents' relation to the actual values and operation

Digital representation of operations

Digital processing must signal an exception when the result is ineffable.

Image showing the physical operations producing not a representation of a third value but an uninterpretable non-value

Prerequisites (3): When the domain is infinite ...

When the domain is infinite, and our representations are finite:
  • There will be values we cannot represent: ineffable values.
  • Such ineffable values may arise as result of operations.
Therefore:
  • We must be able to distinguish between:
    • operations with interpretable results
    • operations with uninterpretable results
  • For example: arithmetic overflow, NaN (not-a-number) values.
Much of any critique of digital reason will consist in teasing out the consequences of these observations and understanding their effects.

Responsibility of consumer

It is not enough for the datatype creator to signal exceptions.
Users of the datatype must pay attention.

An example (NLTK version of Brown Corpus)

A simple example.
  • Original Brown Corpus signals untranscribed formulas with **F.
  • In preparing the NLTK version of Brown, some software did not distinguish words from non-words. So “Af” is the most common noun in English, according to NLTK.

Another example (NLTK Brown)

Original Brown (one sentence) 
E10 0880 ... The .222's have
E10 0890 the short action; the .243 and .308, the medium action, and the .270,
E10 0900 **f and the Magnums, the long action (about $135 for the Standard
E10 0910 Coltsman and $200 for the Custom version). ...
NLTK Brown with POS tags (five sentences):
  • The/at
  • have/hv the/at short/jj action/nn ;/. ;/.
  • the/at
  • and/cc ,/, the/at medium/jj action/nn ,/, and/cc the/at
  • ,/, Af/nn and/cc the/at Magnums/nps ,/, the/at long/jj action/nn ...
Why? Because some software forgot that full stops (.) can occur within ‘words’ like “.222”.

Critique of data structures

Much of any critique of digital reason will be essentially a critique of digital representations (data structures).
  • Some data representations are well understood (integers).
  • Some data representations are exhaustively defined (floating-point numbers).
  • For some data representations, there is helpful preliminary work (XML, JSON, relational data, ...)
  • For some data representations, we must plough new land.

What can we do digitally?

The operations of digital reason

Digital reason and conventional logic

Can proofs surprise us?

Can databases surprise us?

The same arguments apply to collections of data:
  • When querying a database, all you get back out is what you put in.
  • So how can a database possibly help us discover anything new?

How to be surprised

Perhaps we should not be surprised by proofs, or by databases. But we are.
  • Surprise is not a logical concept but a psychological one.
  • Corpus linguists have long since shown that we don't have reliable access to all facts about a language, even if we “know” the language.
  • Machine learning demonstrates that there is information in our data that we do not consciously perceive.
  • Machines see things we do not, because machines never get tired.
  • Machines are particularly good at clerical work, bookkeeping, and similar tasks: e.g. logical proofs.
  • For example: the four-color theorem: 1936 cases to consider.
  • For example: concordances, text searching.

Black boxes, white boxes

We should distinguish two kinds of artificial intelligence work:
  • ‘symbolic’ AI:
    • make the rules explicit
    • make the machine follow those rules
    • requires insight into the rules we follow, and how to follow them
    • often fragile, does not scale well
  • machine learning:
    • less fragile
    • process ‘training data’ to learn rules without human intervention
    • machine representations often opaque to humans
    • reasons for conclusion may be unclear

What do we want?

What do we want?
  • To get things done? Machine learning is fine.
  • To understand things? Machine learning does not help.
  • Scholarship is not the accumulation of facts.
  • Scholarship seeks explanation, understanding.
  • Four-color theorem disappointed mathematicians.
Digital reason is no use to us as scholars, if all it does is produce results.

Accumulation of errors

Another danger: accumulation of errors. Von Neumann wrote:

The characteristic fact regarding these procedures is that when they are broken down into their constituent elements, they turn out to be very long. This holds for all problems that justify the use of a fast computing machine -- i.e. for all that have at least a medium degree of complexity. ... The operations performed in the course of the calculations may amplify errors that were introduced by earlier operations. ... [T]he art of [numerical] computing consists to no small degree of measures to keep this effect down.

In humanities fields, it's hard to quantify errors (and thus hard to limit them).

Accumulation of errors (2)

Example: a panda (57.7% confidence)

Picture of a panda bear with black eyes and white face

Accumulation of errors (3)

Example: a gibbon (i.e. a great ape) (99.3% confidence)

What looks to a human like the identical picture of a panda bear

Responsibility

We are at the dawn of a new world.
  • Machine-learning systems can
    • evaluate loan requests
    • flag tax returns
    • estimate risk of criminal recidivism (re-offense)
    • recommend penal sentences
  • Perhaps such systems can be objective, free of human biases.

Responsibility (2)

But ...
  • Machine-learning systems learn from training data.
  • Training data are typically historical, or manually constructed.
  • Minority populations may will be underrepresented.
  • Historical data may be (will almost certainly be) biased.
  • Even when we exclude sensitive factors (e.g. race), machine learning may learn correlations between variables, to allow it to discriminate in ways that are morally unacceptable and often illegal.

Responsibility (3)

  • If we cannot understand the machine's rationale for its recommendations,
    • we cannot evaluate them critically
    • we cannot take responsibility for agreeing with them
    • we are ceding our human responsibility to someone else
    • the exact opposite of enlightenment
A critique of digital reason must address issues of responsibility: what can we safely delegate to machine learning? How can we retain our autonomy in a world of machine-learning?

Making plans

Kant's critique also touched on sketches like mine:
Plane machen ist mehrmalen eine üppige, prahlerishe Geistesbeschäftigung, dadurch man sich ein Ansehen von schöpferischem Genie gibt, indem man fodert, was man selbst nicht leisten, tadelt, was man doch nicht besser machen kann, und vorschlägt, wovon man selbst nicht weiß, wo es zu finden ist, wiewohl auch nur zum tüchtigen Plane einer allgemeinen Kritik der Vernunft schon etwas mehr gehöret hätte, als man wohl vermuten mag, wenn er nicht bloß, wie gewöhnlich, eine Deklamation frommer Wünsche hätte werden sollen.
Making plans is often a lush, grandiloquent occupation for the mind, through which one assumes an appearance of creative genius by demanding what one cannot oneself do, criticizing what one cannot oneself do one whit better, and proposing what one has no idea how to find, even though somewhat more would have been necessary for a competent plan for a general critique of reason, if it is not to turn out (as usual) a mere declamation of pious hopes.

Conclusion

What must a critique of digital reason achieve?
My pious hopes:
  • A critique of digital representations
  • A critique of conventional logic
  • An understanding of how to retain human autonomy and responsibility

Acknowledgements and details

Thank you

Thank you.
Any questions?