What does Whitehead mean by calling science anti-rational?

The end of the quote you gave says the following:

Science repudiates philosophy. In other words, it has never cared to justify its faith or to explain its meanings; and has remained blandly indifferent to its refutation by Hume.

What Whitehead meant is the following. He imagines that justification is necessary for rationality. Science does not justify its findings. Nor do scientists care that the justification philosophers gave for science, inductivism, was refuted by Hume. See also

Is this inductive or deductive?

Whitehead and the sort of scientists he describes are both wrong on this issue. Justification is impossible and the belief in justification is anti-rational - it ignores the criticisms of the idea of justification:

Do all epistemologies suffer from the "regress of justifications" problem?

Knowledge is actually created by noticing problems, guessing solutions to those problems, and criticising the proposed solutions until only one is left. Scientists use a specific kind of criticism: observations of phenomena for which different theories make different predictions. The lack of justification hasn't held back scientific progress because nobody has ever actually used justification, pretense to the contrary notwithstanding. However, the lack of understanding of how scientific knowledge is created has led to some extremely serious mistakes. In addition, many scientists wrongly believe in justification, and wrongly believe their ideas are justified. This has a terrible cost in terms of scientists sticking to refuted theories on the pretense that they are justified, and recommending coercive imposition of policies allegedly based on their 'justified' ideas.

I'd agree with @mobileink. Science pays no attention to its foundations and endorses unproven metaphysical conjectures. Even if the conjectures happen to be correct this is not a rational or scientific approach.

I would not agree that science has ever shown our 'rationalizations to be wrong' where they are, as it were, rational rationalizations. Just as metaphysics cannot contradict science so science cannot contradict metaphysics.

Science may choose to endorse a naive materialism but it fails in metaphysics, where rationality and logic count for something.

To answer this, its helpful to have some background on Whitehead's understanding of rationalism. In Process and Reality (42) Whitehead defines rationalism as the hope that we can create a general theory, such that everything we find in experience can be held up as an example of that theory. He calls it a "hope" since Whitehead argues there will always be unknowable unknowns that prevent us from completing such a theory. Rationalists are those who nevertheless hold onto this hope, as an "ideal which is seeking satisfaction". In this respect, for Whitehead rationalism is a tipping point where science/philosophy crosses over into religion. It always falls short of being a religion because it remains an ideal, beyond reach, rather than a premise upon which a religious dogma can be formed. According to Whitehead, the Enlightenment is an epoch when this ideal thrived.

Whitehead argues that modern science is anti-rationalist because it has given up this ideal and become self-content and professionalised - he asserts that science increasingly is a faith that is built on weak, enfeebled, superficial or arbitrary starting points and premises. It relies on technicians who blindly following habit, using technical instruments instead of imaginative thinking, mistaking abstractions for concrete reality, and accepting contradictions as good enough.

His books go into this in more detail, drawing out the kinds of paradoxes and contradictions that underly scientific enterprises. For example, he offers this retorsive argument: scientists make experiments in which they rely on final causation (for instance, taking photographs in order to study features of the environment) but then propose models in which there is only efficient causation - without attempting to reconcile the contradiction between what they are doing and saying. Whitehead argues this willingness to ignore the paradoxes of science is anti-rationalist, since it leads science into ruts and blind alleys, instead of encouraging scientists to explore the problem space more widely and imaginatively.

In response to Cort Ammon's statement:

Whitehead's argument is that science as a whole is simply not interested in "justifying its faith or to explain its meanings."

Given this trend is quite strong today, and Whitehead was an author on a then undisputed mathematical approach to proving all true statements in mathematics, it's no surprise that he would happily use such strong words. It's not until Gödel showed that not only did PM not accomplish its goals but also that no such document could ever accomplish them that Whitehead's bubble was popped.

I disagree. Whitehead worked on PM for over ten years with Russell, publishing in 1910, 12 and 13, and then abandoned the project. PM's logicism was therefore unproven and incomplete. It was never 'undisputed'.

It is also false to infer Whitehead was "happily using such strong words" until his "bubble was popped" by Gödel. Gödel's publication was in 1931. Russell and Whitehead worked on PM twenty years earlier, and Whitehead had long since moved onto other projects. Whitehead indirectly acknowledges Gödel’s proof in 1938, writing: "Today, even Logic itself is struggling with the discovery embodied in a formal proof, that every finite set of premises must indicate notions which are excluded from its direct purview." (MT 2). Whitehead immediately adds that philosophy "should never start from systematization." He credits this to William James, whose "intellectual life was one protest against the dismissal of experience in the interest of system. He had discovered intuitively the great truth with which modern logic is now wrestling." (MT 3). I believe that, doing PM, Whitehead already intuited this great truth - the shortcomings of logical formalism. Reading Whitehead's Process and Reality (1929) confirms this. e.g. Whitehead writes that systems "do not exhibit mere illogicalities. They suffer from inadequacy and incoherence" (6) - For Whitehead, Gödel's proof was redundant... but nice to have.

PS. For those interested in Whitehead, The Quantum of Explanation by Auxier and Herstein is a great read.

All he's really saying is that science prefers evidence to argument. By "anti-rational" he does not mean irrational. He means something like "against the idea that we should look to our own reason as the ultimate source of scientific authority". Which is a little odd considering how frequently science shows our rationalizations to be wrong.

But I don't think he's really talking about working scientists. I think his target is the philosophical "foundations" of science, in which case he's right. Working science doesn't general care about philosophical foundations, as far as I can tell.

At the start of the chapter of Science and the Modern World where he turns back toward philosophy proper, he comes close to a real characterization, of what he means by rationalism, though not of rational:

Philosophers are rationalists. The are seeking to go behind stubborn and irreducible facts: they wish to explain in the light of universal principles the mutual reference between the various details entering into the flux of things. Also, they seek such principles as will eliminate mere arbitrariness; so the remainder of things shall satisfy some demand of rationality. They demand meaning.

So in some sense, complete coverage without compromise is the goal, and that includes compromise with the arbitrariness of terms and the fitting of concepts into narrow molds that has historically been required by science of itself. (This is especially true now that science itself has found those molds do not accommodate it.)

(To your point about Greeks: Whitehead takes the position very early in that same chapter that we cannot define philosophy in terms of antiquity. We simply no longer have access the same thought processes as ancient people, we lack the context to actually recover them, and can only address their writings from our own perspective, which is inescapably different.

He invests some time in addressing the notion that science is overly rationalistic, blaming the misinterpretation on the Romantic Movement.)

Science does not buy into this agenda wholeheartedly, for the same reason Protestantism and the counterreformation keep Western religion from really doing so anymore or ever again: because it is, at root, based in a faith that cannot be rationally justified, and it does not care. To that degree, it is antipathetic towards rationalism in this sense.

Pressed for philosophical justification for having faith in induction or statistical inference, scientists as a whole simply fall back on success as proof. Success is not justification, it is simply success.

When that is criticised deeply enough they back off to claiming that science is not an attempt to explain, but only to predict, because then success would be all that matters. Yet we use it to explain. We do not simply amass data that we then draw correlations out of, we make models and presume that relatively few verifications of the model make it reliable, just because the model itself has an intrinsic mathematical structure.

If the goal were simply to predict, we would leave it at raw statistics and not contrive theoretical mechanisms. And when we did use theories, we would not think of them as intrinsically meaningful, which we most certainly do. They would have to be simply shorthand summaries of the observed correlations.

At its base, faith in statistical inference simply has to be taken as a fact, and it cannot be rationalised. We need to believe that our experience of the regularity of past interactions with nature means it is basically regular, and that it somehow actively resists 'miracles'. Otherwise, we could equally explain success by being on the right side of God, just very lucky, or deluded. But no real proof of that assertion is possible.

He is contrasting axiomatic, positivist thinking with empiricist, scientific thinking. Science is anti-rational in the sense that it rejects pure reason as a means of understanding the world. Science uses empirical evidence based on unprovable assumptions such as the accuracy of the measuring apparatus. Furthermore, climatology, evolutionary biology, and most of what is called "science" today completely rejects axiomatic reasoning and instead substitutes graphs of vector auto regression models developed by grad students at 2 in the morning using Matlab as the fundamental means of understanding natural phenomena.

Science and the Modern World was first published in 1925. This is 12 years after Whitehead and Russel published Principia Mathematica but 6 years before Gödel poked holes in it with his incompleteness theorem.

Science has always encouraged the axiomatic belief that the scientific method leads to knowledge. It has a great track record, but it is not possible to prove it within the confines of science without having to resort to circular logic (one can look at the work of Popper and the refutation that followed his work for excellent arguments on both sides of the debate). There are indeed scientists who seem aware of this, but Whitehead's argument is that science as a whole is simply not interested in "justifying its faith or to explain its meanings."

Given this trend is quite strong today, and Whitehead was an author on a then undisputed mathematical approach to proving all true statements in mathematics, it's no surprise that he would happily use such strong words. It's not until Gödel showed that not only did PM not accomplish its goals but also that no such document could ever accomplish them that Whitehead's bubble was popped.

I think this means that he considers that science lacks metaphysical basis. Scientific pretends that metaphysic is irrelevant to their work, but by ignoring this subject, they risk to use naive metaphysical assumptions that affect their progress.

He was right to critic naive materialism. However, scientifics had already demonstrated that they were able to question naive assumptions about nature of the universe, as quantum mechanics and general relativity demonstrates.