Tag: Cambridge

We Are Finite. Does This Affect What We Can Know?

§2. What Can We Know?

The theme of this section is that We Are Finite: while our knowledge of the number 3 is intimately associated with experiences of some kind or other of triples of one or another sort, surely nothing like that is the case for our knowledge of the number ω or, for that matter, the number 10101010.

To avoid misunderstanding, let me emphasize that the issue being addressed is not “How can we have knowledge of mathematical objects despite their abstractness?” but rather “How can we have knowledge of infinite mathematical objects despite their huge size and consequent remoteness from experience?” Here is what I mean by remoteness: Not only do we not have experience of infinite mathematical objects, but we do not have any experience of anything suitably like them. The number 2 bears some relation to pairs of objects. The points and lines of Euclidean geometry bear some relation to pencil points and lines. But there does not seem to be anything suitable to play any analogous role for infinite mathematical objects.

Of course infinite mathematical objects are abstract. The issue of remoteness is an addition to that of abstractness. Both are apparent epistemological difficulties caused by the distance of mathematical objects from experience.

But if we divide the problem of the abstract into two parts, the problem of the finite and the problem of the infinite, it becomes clear that the two have quite distinct features.

The problem of the abstract, in Paul Benacerraf’s words, is this [Ben73, p. 409]: “the concept of mathematical truth, as explicated, must fit into an over-all account of knowledge in a way that makes it intelligible how we have the mathematical knowledge that we have. An acceptable semantics for mathematics must fit an acceptable epistemology?” For Benacerraf, an acceptable semantics is a Platonist one, and so his problem is that of the difficulties involved in giving an acceptable account of knowledge of abstract objects. Benacerraf concentrated on a weaker version of the problem: not on accounting for the mathematical knowledge that we have but on accounting for how we can have any mathematical knowledge whatever. “The minimal requirement, then, is that a satisfactory account of mathematical truth must be consistent with the possibility that some such truths be knowable” [Ben73, p. 409]. Philosophers have tended to concentrate on the minimal requirement and to offer up solutions to the problem of the finite to solve it.

Even those skeptical about the existence of any abstract mathematical objects at all want to endorse the truism ‘2 + 2 = 4’ in some way or other that acknowledges that it is better than ‘2 + 2 = 5’ even though they may not grant that it is true. Even without an abstract number 2, one is still faced with explaining the general fact that the members of two nonoverlapping pairs form a quadruple.

Whatever one’s views about mathematical objects, it is necessary to make sense of our counting, computing, and bookkeeping activities. Skepticism about small finite mathematical objects—in particular small natural numbers is just not doubt about the acceptability in some form or other of many of the putative facts about them. Moreover, there are many stories one could tell about a source for genuine knowledge concerning some finite mathematical objects, including, as the most trivial special case, knowledge of their existence. Take, for example, small natural numbers. Various explanations of our knowledge about them might invoke the experience of time, the experience of bunches of physical objects or of patterns exhibited by them, or the sequencing of words in sentences. Other explanations might rely on the exigencies of the construction of theories of the physical world. We are faced with too many ways of accounting for our knowledge of small finite mathematical objects, not too few. For example, Parsons [Par80] showed how to account for such knowledge on the basis of our linguistic capacities, while Maddy [Mad90] showed how to do it on the basis of experiences with medium-sized physical objects.

Benacerraf’s minimal requirement can surely be met with an acceptable solution to the problem of the finite—though I make no claim to know what the actual solution is. That is a question whose answer involves detailed psychological information about how people typically actually acquire knowledge of small finite mathematical objects, and we do not yet have sufficiently detailed information to answer it. Acquisition of the number concept does, however, involve both linguistic components—learning to count aloud—and experience of medium-sized physical objects—counting them using the spoken number sequence. It is therefore likely that the actual solution involves components of both the one proposed by Parsons and the one proposed by Maddy—and probably other components as well.

In sharp contrast to the situation about ‘2 + 2 = 4’, many of those who are skeptical about the existence of infinite combinatorial collections would want to doubt or deny the Axiom of Choice—not only its truth, but its acceptability in any form whatever. General facts about the infinite are not robust in the same way that the facts of counting, computing, and bookkeeping are. Moreover, it is not at all clear what we can fall back on as a source of mathematical knowledge concerning the infinite—what can play the role that bunches and sequences of moments, objects, or words seem so well suited to play for small finite mathematical objects. It is that lack that raises the problem posed by the remoteness of the infinite: it seems that we cannot have grounds to know what we find we actually do know about the infinite.

In Chapter VIII I shall show that as a matter of fact the combinatorial infinite is not remote—it has pretty much the same kinds of ties to experience as do small natural numbers. (That may be a bit misleading see Chapter VIII for a more careful formulation.) That solves the problem of the remoteness of the infinite philosophical problems concerning infinite mathematical objects become just like the familiar ones concerning finite mathematical objects. That is important because the problems concerning finite mathematical objects are not skeptical ones—the genuine doubts about the acceptability of our theory of the infinite are refuted. It also provides the essential missing ingredient for an explanation of the grounds on which mathematicians are entitled to make claims of self-evidence. But before presenting the solution, it is necessary to become clearer on the nature of the problem.

The two Benacerrafian problems—of the finite and the infinite—are both important, and every adequate philosophy of mathematics must be compatible with solutions to them. Nonetheless, the problem of the infinite deserves special emphasis because it is in danger of being lost as the result of the huge amount of attention being devoted to the Benacerrafian problem of the abstract in its simplified guise as the problem of the finite. That loss would be most unfortunate. The problem concerning the infinite was a primary concern of the philosophy of mathematics for many years—as may be seen in the work of Brouwer and that of several philosophers discussed by Benacerraf, such as Hilbert, Gödel, and Quine.

Shaughan Lavine, Understanding the Infinite, Harvard University Press, 1994, pgs. 162-165.

You do not doubt that ‘7 – 7 = 0’, but you cannot say ‘∞ – ∞ = 0’. In math lingo, the latter is indeterminate. Does this mean we are incapable of understanding the indeterminate, or is this just the nature of the universe? Consider Eugene Wigner’s lecture, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences”. He noted, “It is important to point out that the mathematical formulation of the physicist’s often crude experience leads in an uncanny number of cases to an amazingly accurate description of a large class of phenomena.”

Still, complete understanding often seems to elude us.

World-Watching: The Problem with the Current Russia Sanctions Regime

[from Project Syndicate, by Mohamed A. El-Erian]

There is much debate about the effectiveness of Western sanctions, the Ukraine war’s implications for markets and the global economy, and what the West’s next steps should be. While there are few good options, some are clearly worse than others.

Cambridge — It has been five months since Europe and the United States imposed tough economic and financial sanctions on Russia, a G20 country that was the world’s eleventh-largest economy on the eve of its invasion of Ukraine. While the sanctions have been gradually strengthened in the intervening months, debate rages about their effectiveness, the war’s broader implications for markets and the global economy, and what the West’s next steps should be.

On the first question, although the sanctions have been less effective than Europe and the U.S. had hoped, they also are proving more onerous than the Kremlin claims. Russia’s central bank expects GDP to contract by 8-10% this year, while other forecasters expect a larger fall, together with longer-lasting damage to growth potential. Imports and exports have been severely disrupted, and inflows of foreign investment have essentially stopped. Shortages are multiplying, pushing inflation higher. At this point, the country no longer has a properly functioning foreign-exchange market.

The sanctions would have bitten much harder had the West not opted for a carve-out of Russia’s energy sector, and had many more countries joined the U.S. and Europe in the effort. Because that didn’t happen, Russia has not felt nearly as much pressure as it would have. Moreover, it has been able to continue trading through various side and back doors that will likely become increasingly important as long as the sanctions regime, as currently designed, continues.

Nonetheless, it is only a matter of time before the Russian economy experiences a harder hit. Inventories of imported goods – including many critical technological and industrial inputs – are dwindling fast, and many sectors are becoming less resilient. The cumulative damage to Russia’s economy over time will be significant and long-lasting – a fact that has not yet been fully captured by consensus medium-term forecasts.

The second question concerns global spillovers from the war and the sanctions regime. Most observers agree that Russia’s invasion has increased not just energy insecurity but also food insecurity, highlighting the fallout from the war’s disruption to Ukrainian agricultural exports. But there is still much debate about the West’s use of the economic nuclear sanctions option: the curbs placed on Russia’s central bank and on Russia’s use of the international payments system.

These curbs are far more intrusive than the usual mix of restrictions on sanctioned government and private sector trade and on individuals’ financial dealings. Yet, because they are not subject to any internationally agreed standards, guidelines, or checks and balances, they fall outside the purview of relevant global-governance bodies such as the Bank for International Settlements, the International Monetary Fund, and the World Trade Organization.

In a time of war, such oversight might seem like a nicety. But some worry that the sanctions could significantly reduce the dollar’s role as the world’s reserve currency and the U.S. financial system’s role as the primary global intermediary for other countries’ savings and investments. After all, a growing number of countries undoubtedly now feel more vulnerable to the reach of U.S. sanctions.

But it is impossible to replace something with nothing, which means that no significant loss of dollar or U.S. financial primacy will occur in the immediate future. Rather, the sanctions will lend further momentum to the gradual process of global economic fragmentation, which was also fueled a few years ago by the tariffs imposed by the Trump administration. More countries now have even more of a reason to pursue greater financial resilience and inherently inefficient forms of self-insurance.

That brings us to the third debate. With no end in sight for the war, what should the West do next? Fearing the implications for energy prices and the supply of gas to Europe, many in the West are tempted to call for a moratorium on any new sanctions – or even for additional carve-outs. Others, however, favor additional measures to hold Russia accountable for its indiscriminate attacks on Ukrainian civilians.

In any case, maintaining the current sanctions regime is not problem-free, owing to the twin objectives of pressuring Russia and limiting the economic disruption to Europe. Moreover, as European Commission President Ursula von der Leyen recently said, it feels as if Russia is “blackmailing” Europe by threatening to disrupt gas supplies at any moment. No wonder the Commission is urging member countries to cut consumption by 15%.

Under the current sanctions regime, the West risks falling between two horses. While easing sanctions could help alleviate concerns about Europe’s economic outlook, this option is a non-starter, given the atrocities that Russian forces are committing in Ukraine. But if the West is serious about pressuring Russia through truly crippling economic and financial sanctions, it needs to bite the bullet and eliminate the carve-outs for energy.

Doing so would undoubtedly have a severe short-term economic impact on European economies and the rest of the world, amplifying the “little fires everywhere” syndrome that I warned about in May. It is therefore critical that governments use their available fiscal space to provide targeted support to vulnerable segments of the population, as well as to fragile countries; and multilateral agencies must support developing countries through aid and a more operational debt relief framework. If done properly, this option would yield better outcomes in the medium and long term than the current strategy.

Muddling through risks bringing about the worst of all possible worlds. It is insufficient to dissuade Russia from continuing its illegal war; it is fueling deeper fragmentation of the international monetary system; and it is not even protecting Europe from a winter gas disruption.

Mohamed A. El-Erian, President of Queens’ College at the University of Cambridge, is a professor at the Wharton School of the University of Pennsylvania and the author of The Only Game in Town: Central Banks, Instability, and Avoiding the Next Collapse (Random House, 2016).