Thinkers have naturally questioned whether the phenomena of uncertainty can be usefully handled by the quasi-mathematical notions of `probability.' Certain subsets of uncertainty those dealing with risks, gambling, insurance, repetitive inventory and quality control of production, and even with tactics of repeated investing are thought to lend themselves better to useful employment of probability procedures.--PA Samuelson Foundations of Economic Analysis, (1983) enlarged edition [pp. 503-504].
By anti-mathematicism I characterize expressed reservations about the high prestige, utility, and (either/or) epistemic authority of applications of mathematical techniques in (some areas of) philosophy and the sciences. The idea presupposes a distinction between pure and applied mathematics that I will not question for present purposes. Here's a canonical statement of the idea (from Hume): "in all demonstrative sciences the rules are certain and infallible; but when we apply them, our fallible and uncertain faculties are very apt to depart from them, and fall into error." (Treatise 184.108.40.206)
The first edition (1947) of Samuelson's Foundations self-consciously ushered in the formal revolution in economics. In it, he assumed "a world involving no transaction friction and no uncertainty.” (123.) This helped facilitate what I call a whole-sale displacement strategy in which (epistemic and metaphysical) uncertainty just is treated as measurable risk within economics. This strategy simply denied the validity of a distinction widely embraced by an older generation of economists (including Knight and Keynes,) between unmeasurable uncertainty and measurable risk. (As I noted, Arrow did much of the heavy-lifting against the distinction, but the strategy does not originate with Arrow.)
The passage quoted above from the appendix to the enlarged edition of Samuelson's Foundations [HT MA Khan] is more measured, even ironic (it's ironic because Samuelson had treated gambling and repeat investment jointly, see here). In it Samuelson deploys what one might call a containment strategy in which the successful application of mathematical technique is restricted only to some (limited) domain (that is, "gambling, insurance, repetitive inventory and quality control of production, and even with tactics of repeated investing"); it does not include the whole of economic life. Moreover, Samuelson (who is a very careful writer), treats the nature and scope of this containment strategy itself not as a settled fact, but as a matter of dispute or, to be more precise, -- Samuelson knew his Quine -- opaque context ("are thought").
That is to say, Samuelson recognizes that the deployment of probability procedures in models of a sub-set of reality ("gambling, insurance, repetitive inventory and quality control of production, and even with tactics of repeated investing") should be treated with caution. (One wonders how Samuelson, who died in 2009, responded to the crisis of 2008 in which probabilistic risk-models fared badly.) One is not permitted to treat the modal universe of these models as identical to those of wider economic reality.
Of course, Samuelson's containment strategy does not rule out the application of other formal techniques in wide domains of economic life beyond "gambling, insurance, repetitive inventory and quality control of production, and... tactics of repeated investing." So, the passage should not be read as a limitation on the formal revolution. But it is an acknowledgment that reliably connecting decision-rule-generating-mathematical-economics to bits of economic reality is harder than he may have believed when he was a young revolutionary.