Double-entry bookkeeping and Galileo: abstraction vs idealization

Two weeks ago, I wrote a post on how abstract is not the opposite of empirical. In that post, I distinguished between the colloquial meaning of abstract and the ‘true’ meaning used by computer scientists. For me, abstraction is defined by multiple realizability. An abstract object can have many implementations. The concrete objects that implement an abstraction might differ from each other in various — potentially drastic — ways but if the implementations are ‘correct’ then the ways in which they differ are irrelevant to the conclusions drawn from the abstraction.

I contrasted this comp sci view with a colloquial sense that I attributed to David Basanta. I said this colloquial sense was just that an abstract model is ‘less detailed’.

In hindsight, I think this colloquial sense was a straw-man and doesn’t do justice to David’s view. It isn’t ignoring any detail that makes something colloquially abstract. Rather, it is ignoring ‘the right sort of’ detail in the ‘right sort of way’. It is about making an idealization meant to arrive at some essence of a (class of) object(s) or a process. And this idealization view of abstraction has a long pedigree.

In this post, I want to provide a semi-historical discussion of the the difference between (comp sci) abstraction vs idealization. I will focus on double-entry bookkeeping as a motivation. Now, this might not seem relevant to science, but for Galileo it was relevant. He expressed his views on (proto-)scientific abstraction by analogy to bookkeeping. And in expressing his view, he covered both abstraction and idealization. In the process, he introduced both good ideas and bad ones. They remain with us today.

Let me start at the beginning.

Mathematics started with representation. If we look at the earliest forms of language, they were used for record keeping. Recording the number of goats or cows or bags of grain. This required seeing certain symbols of record as representing physical objects. And this seems to be conceptually independent of or prior to writing more generally, at least if we take the Inca as a guide. This representation required a certain amount of abstraction. Three cows could be Daisy, Clementine, and Spots or Old Reliable, Salty, and Beatrice. The particular cows did not matter for the accuracy of the record and thus, the numeric record abstracted over the particular cows. But they were still all cows.

As most good ideas in Latin Europe, abstraction really took off after appropriating ideas from the Muslim world. In this case, an idea central to commerce was absorbed by Genoa, Venice and Florence from the Jewish merchants in Old Cairo[1]: double-entry bookkeeping. We see the first extant records of Italian double-entry bookkeeping from around the late 12th and early 13th centuries, with Amatino Manucci’s 1299-1300 records for a Florentine merchant partnership based in Nimes and the 1340 Messari accounts for the Republic of Genoa. By 1494, Luca Pacioli had summarized and codified these Northern Italian practices in a section of a chapter on business in his mathematics book Summa de arithmetica. This was the only comprehensive mathematics textbook at the time and became a standard reference text throughout the 16th century and beyond.

The standard system of double-entry bookkeeping in the Renaissance consisted of three books that proceeded up levels of abstraction. In the first book, you recorded everything exactly as it happened with as much detail as possible. You might write that “On June 14th at the main market square, I received a good cow Daisy from Giovani in exchange for ten soldi”. In the second book, this was reduced to a list of transactions. Maybe you would have a column for date (June 14th), type of transaction (bought cow), and price (10 soldi). Finally, the third book would be the account book. On facing pages you would have a list of credit (Latin for “he trusts”) and debit (Latin for “he owes”), with every transaction appearing on both pages. In this case, you might credit 10 soldi to your current account and debit 10 soldi to your assets. By the time you reached this book, everything has been translated into notional cash value. Cows, debts, paintings, and property were now represented just by a price. Even if you do not know if you will ever sell the asset or what you will get for it if you do. The numbers were no longer just cows. An entry in a list of credits and debits could be implemented by a variety of physical objects (cows, paintings), social and governmental obligations (debts, stakes in companies), or something in between (property).

For me, this reflects a very powerful notion of abstraction via multiple realizability. And this importance was not lost on the mathematicians and proto-scientists of the day. Although it was mixed in with an idealization view of abstraction.

By the 16th century, one of the best ways for mathematicians it Italy to make money was by teaching this standard system of double-entry bookkeeping. One particularly notable Italian mathematician who had to make his money this way before getting a university position was Galileo. In 1585, Galileo finished his university studies and he did not secure a university appointment until 1589. His most likely source of income for those four years was probably as a teacher of accounting, which included double-entry bookkeeping.

Galileo wrote explicitly on what he saw as the role of abstraction in science. When responding to challenges that his law of falling bodies did not correspond to the real world, he used an analogy to bookkeeping. I want to go through his analogy in three parts.

Galileo started:

[w]hat happens in the concrete … happens in the abstract. It would be novel indeed if computations and ratios made in abstract numbers should not thereafter correspond to concrete gold and silver coins and merchandise …

Here Galileo is using the multiple realizability view of abstraction. The fact that you can make computations with abstract numbers without needing to directly reference the particular objects that implement them. In particular, given a proper abstraction, one can make an inference or calculation at the level of numbers with results that would then correspond to some specific physical effect or process. This was revolutionary for Italian commerce, since it allowed a better accounting of profits and the ability to form more robust partnerships between non-family.

So far, so good. But this is not enough to save Galileo. For the challenge he faced was on why he ignored air resistance when reasoning about falling objects. To address this, he has to implicitly shift from abstraction to idealization:

Just as the bookkeeper who wants his calculations to deal with sugar, silk, and wool must discount the boxes, bales, and other packings, so the mathematical scientist, when he wants to recognize in the concrete the effects which he has proved in the abstract, must deduct the material hindrances, and if he is able to do so, I assure you that things are in no less agreement than arithmetical computations.

Here we can see that David is in good company in his colloquial interpretation of abstraction. For my view, Galileo makes the same mistake in taking idealization as abstraction.[2] Here the detail of boxes, bales, and other packing is removed as irrelevant and not as multiply realizable. I think this is easier for the modern reader to see in a subject closer to what Galileo was actually defending instead of in the book-keeping analogy. Let’s look at the frictionless plane[3].

Consider a statement like: “pushing and then letting go of an object on a plane will result in that object continuing its motion at a constant speed relative to the plane.” This statement abstracts over objects: it doesn’t matter if we push a box of sugar, a bale of wool, or even Daisy or Spots. However, it is not abstracting over friction. This is because if friction was present then the statement would be false: the object would stop moving. Instead, we are idealizing. We don’t believe that frictionless planes exist in our physical world, but we think it is good to imagine them as an intuition pump and as a way to factor our different aspects of motion. It lets us get to — or define — a certain essential feature. This is extremely useful and important. But for me, it is not abstraction.

Galileo can now mount his final defense, and the most dangerous part of the argument:

The errors, then, lie not in the abstractness or concreteness, not in geometry or physics, but in a calculator who does not know how to make a true accounting.

Here, Galileo is doing both a good and a bad thing. The good aspect — for the development of physics, at least[4] — is that he is reafying the idealization. He is saying that the ideal frictionless plane is not just a mathematical trick, but a potential physical object. It is just that in practice this real physical aspect is muddied up by friction. Probably all because Adam got us kicked out of Eden. At the time that Galileo was making this statement, it was revolutionary. Before, mathematics was not seen as relevant to physics. Thus, it was important to assert the potential reality of the idealized object and thus link mathematics to the world. Something that might have seemed strange to Aristotle — or at least his medieval interpreters.

The bad aspect of Galileo’s closing is that he has turned idealization into a crutch. If you don’t observe the results that he predicts, it is not the fault of his “true accounting” but the fault of the messy extra factors. Even more restrictively: if you calculate results that are very wrong, it is because you ‘ignored’ the wrong aspects of the problem. Thus, a person who works with idealization needs to develop an artistic sense for what aspects can be idealized away and which cannot. Which effects are ‘small’ or ‘not central’. In the case of physics, there is often agreement on what should be idealized away. And one of the difficulties that physics students face is in learning these community standards on what to ignore and when.

But in fields like mathematical oncology, there is much less consensus on what parts can be idealized away and which parts cannot. And people who disagree can end up talking past each other. Or, ‘resolving’ their disagreement with kitchen sink models that throw away the whole point of idealization. That is why in oncology (and biology more general), it is better to abstract rather than idealize. To guarantee our conclusions regardless of the details of the implementation, instead of relying on our intuition about which details can be idealized away for simpler analysis.

In shorter words: simple idealized models are good but abstract models are better.

Notes

  1. Double-entry bookkeeping could have also been learned by the Italians from other majors Eastern Mediterranean trade cities like Constantinople or Alexandria. However, we have explicit examples of 11th century records from Old Cairo, so it is a good historic anchor point. It is also import to note that the system was likely developed independently by merchants in other parts of the great silk road. For example, a similar four-element bookkeeping system originated in the important Korean trade city of Kaesong sometime in the 11th or 12th century. This suggests to me that the ideas of double-entry bookkeeping were certainly ‘in the air’ throughout the silk road. I’d be interested to read more history on this, if anybody has recommendations.
  2. For the quote, I am relying on pg. 164 of David Wootton’s “The Invention of Science” (Penguin Books, 2016). Given that Galileo did not write in English and — more importantly — that terms drift over time, it is unreasonable to say that Galileo is misusing the term ‘abstraction’ in his historic context. For that, I would need to first establish how that word was used during his age. However, this would just by a discussion of a colloquial use, and my point is not to suggest that Galileo (or David) are using the term incorrectly in the colloquial sense. Rather, it is to show that the colloquial sense of abstraction actually has two distinct — but related — concepts within it: what I call abstraction and idealization.
  3. The friction-less plane would have been new to Galileo’s contemporaries, since he is the one that introduced the idealization in 1608. Hence it couldn’t be used as an anchor for the analogy. More importantly, the analogy I quote was in the context of explaining why friction can be ignored, so it’d be silly to reference another case of friction being ignored — it’d be just as contentious. But I am not trying to defend Galileo’s physics here, just to illustrate the use of abstraction and idealization.
  4. Reafying heuristic models can be bad outside of physics, however. The most important example might be when the ‘objectivity’ of numbers is used to power weapons of math destruction. An example closer to mathematical modeling in biology might be replicator dynamics and well-mixed models taken as the ‘default’ behavior. I’ve explicitly written against this in my recent BBS commentary with Tom Shultz:

    for evolutionary games, we believe that activity is required to break correlation rather than to create or maintain it. The classic example would be spatial structure, which is known to facilitate cooperation. Although uncorrelated inviscid models are easier to analyze (and were thus analyzed first), they are not more natural because humans (like all other organisms) are embedded in space. To undo this inherent correlation, we need the quintessential active process of locomotion for de-correlation.

    In other words, space is not an aberration to an essentially aspatial world of evolution. Rather, the only reason to consider the aspatial case is due to tractability — which might be enough for effective measures — and not due to some claim to capturing an important aspect of reality.

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About Artem Kaznatcheev
From the Department of Computer Science at Oxford University and Department of Translational Hematology & Oncology Research at Cleveland Clinic, I marvel at the world through algorithmic lenses. My mind is drawn to evolutionary dynamics, theoretical computer science, mathematical oncology, computational learning theory, and philosophy of science. Previously I was at the Department of Integrated Mathematical Oncology at Moffitt Cancer Center, and the School of Computer Science and Department of Psychology at McGill University. In a past life, I worried about quantum queries at the Institute for Quantum Computing and Department of Combinatorics & Optimization at University of Waterloo and as a visitor to the Centre for Quantum Technologies at National University of Singapore. Meander with me on Google+ and Twitter.

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