The Philosophy of Cosmology

Over the Summer I gave a talk to my department about the Philosophy of Cosmology. I had been thinking about doing it for at least a year and even started putting together slides before I committed to giving the talk - which I was hesitant to do, since philosophy, or anything that isn't research, is not exactly encouraged...

Anyway, it ended up being a success! Many more people attended than I thought would care, there was active discussion, it went 30 minutes over time and no one complained, and I generally felt pretty good about it. Since I put a lot of time into reading, thinking, and writing comprehensive notes, I thought I would share them here.

The talk was split into three main parts:

I. Why care about philosophy? (And what's with the haters?)
II. Intro to the Philosophy of Science (Phil Sci 101)
III. Topics in the Philosophy of Cosmology

Most of the purpose of my talk was to educate fellow scientists about philosophy and to convince them they shouldn't be "philosophy haters", so the first two parts (including discussion) took an hour. I don't know if I convinced people, but there were philosophical discussions in the hallways after, so I count the experience as a success!

I. Why Care?

What is Philosophy?

Since much of philosophy involves defining terms, let's start with a definition of philosophy.

Broadly speaking, it can be thought of as an activity that uses reason to explore issues that include the nature of reality (metaphysics), the structure of rational thinking (logic), the limits of our understanding (epistemology), the meaning implied by our thoughts (philosophy of language), the nature of the moral good (ethics), the nature of beauty (aesthetics), and the inner workings of other disciplines (philosophy of science, philosophy of history, and a variety of other “philosophies of”).

~ Massimo Pigliucci [emphasis added]

You could say that philosophy is just thinking about things, but of course it’s more involved… This quote is taken from Massimo Pigliucci (philosopher, evolutionary biologist, and prolific blogger), who blogs a lot about philosophy and science. I learn interesting things from his blog, including whenever a scientist (usually physicist) says something negative about philosophy.

Philosophy Haters

The first of these is Steven Weinberg. In his book "Dreams of a Final Theory", the final chapter is called "Against Philosophy", which starts:

Physicists get so much help from subjective and often vague aesthetic judgments that it might be expected that we would be helped also by philosophy, out of which after all our science evolved. Can philosophy give us any guidance toward a final theory?

(Judging by the title of the chapter, you can guess that he goes on to answer: No.) I could go through like a philosopher and poke holes in the rest of the chapter, finding weak points in his essentially philosophical argument against philosophy, but I just want to point out the emphasized text which I think points to his true problem with philosophy, which the rest of the philosophy haters share, which is that philosophy is NOT SCIENCE. Science has surpassed philosophy, therefore philosophy is no longer needed.

The next hater is Stephen Hawking, and this is his famous "philosophy is dead" quote:

Most of us don't worry about these questions most of the time. But almost all of us must sometimes wonder: Why are we here? Where do we come from? Traditionally, these are questions for philosophy, but philosophy is dead. Philosophers have not kept up with modern developments in science. Particularly physics.

Again his problem is that philosophy is now being surpassed by physics. He seems to be claiming that philosophy can't answer these questions, but science can, with which I wholeheartedly disagree. [More of me disagreeing with Hawking can be found here.]

Lawrence Krauss wrote a book called "A Universe from Nothing", and some philosophers pointed out that his book is not about what it says it's about, and that it doesn't answer the fundamental question of "why is there something rather than nothing" which it claims to. So he responds:

Every time there's a leap in physics, it encroaches on these areas that philosophers have carefully sequestered away to themselves…. And the worst part of philosophy is the philosophy of science; the only people, as far as I can tell, that read work by philosophers of science are other philosophers of science. It has no impact on physics what so ever.

Lawrence also thinks that science is taking over territory which used to belong to philosophers. (And of course, the only people who read papers in theoretical physics are theoretical physicists, so that is not much of a criticism.)

Even Neil deGrasse Tyson gets in on the philosophy hate:

But, philosophy has basically parted ways from the frontier of the physical sciences, when there was a day when they were one and the same. Isaac Newton was a natural philosopher, the word physicist didn’t even exist in any important way back then… It’s not that there can’t be other philosophical subjects, there is religious philosophy, and ethical philosophy, and political philosophy, plenty of stuff for the philosophers to do, but the frontier of the physical sciences does not appear to be among them.

So philosophy and science used to be the same, and now they are not, and philosophers are not doing science anymore... therefore philosophy is worthless?

Maybe we need to remind these haters of what philosophy is (e.g. Massimo's quote above). Actually, let's make it simpler... What is philosophy?

NOT SCIENCE.

All these haters are making a category mistake (which is something you can learn about in philosophy), essentially criticizing philosophy because it isn't science... But it does not follow that philosophy is worthless unless you think that only science is worth doing - but these haters aren't criticizing the history of science because historians are failing to make contributions to the forefront of physical science. [And why is it usually physicists? I could definitely speculate on the psychology of physicists that makes them so susceptible to philosophy hate...]

Philosophical Questions

Physicists are free to think that philosophy is worthless, just as they may about medieval literature... philosophy doesn't concern itself with being useful to scientists. Nevertheless, I think there are some philosophical questions that scientists should care about. So one day I decided to write down whatever came into my head.

The first was, "What counts as scientific explanation?" As scientists, we are in the business of producing scientific explanation, so it's worth it to spend some time thinking about what that means.

Then I thought, "What requires scientific explanation?" [This may have been around the time of the eclipse when I was thinking about whether (and what type of) coincidences require explanation.] This is another important question for scientists to ponder; figuring out what to explain is what will lead you to papers.

A follow-up to that, or kind of an example, is: "Do the values of the physical constants need to be predicted by some (as yet undiscovered) fundamental theory?" [I asked my audience of scientists to raise hands, and if I remember correctly more voted yes than no.] To answer no to this question means that there can be some numbers that will never be predicted but that we need to measure.

Then I realized I phrased the question in such a way as to imply that we discover theories, but that assumes that theories exist out there somewhere already, and maybe I don't want to subscribe to that level of realism about theories... so another question is, "Do we discover or invent theories?" Another follow-up question is, "What properties are necessary for a theory to have?" which we will get into a little bit later.

Finally, "What is the nature of physical law? Does it exist independently of the objects it describes?" This may not be that necessary for scientists to think about, but I personally find it interesting, and we will get to that too on the subject of Realism.

II. Phil Sci 101

The Demarcation Problem

The demarcation problem asks the question: what is it that distinguishes between science and non-science (or pseudoscience, metaphysics)?

Logical Positivism was an attempt to develop a criterion for demarcation in terms of verifiability: a scientific statement is only meaningful if we have some means of verifying its truth (and if not, it's not scientific). There are two main problems with this approach, and the first is that the verifiability condition fails its own standard of meaning! If the verifiability criterion itself isn't meaningful, where are we?

The second problem is more interesting, and it's called the confirmation paradox (due to Hempel). Imagine you have the proposition: All ravens are black. In order to test this proposition, you can collect observations of ravens. Finding black ravens supports your proposition, and finding one white raven would disprove it. However, All ravens are black is logically equivalent to All non-black things are not ravens. This means that, for example, collecting observations of red apples also counts as evidence for the proposition All ravens are black. This is contradictory to intuition; it seems foolish that objects completely unrelated to the subject of the proposition count as evidence for it.

So maybe verification is not a satisfactory principle with which to define science. Falsificationism (due to Popper) instead attempts to distinguish science from pseudoscience by the criterion of falsification: scientific statements must be capable of conflicting with possible or conceivable observations, otherwise they are not scientific. This has some advantages over positivism because it may not be possible to confirm a proposition - for example, it would take observing ALL RAVENS to confirm that they are all black - but it is usually easier to falsify it - it only takes one observation of a white raven.

There are some issues with this too: the first is logical vs. practical falsifiability. For example, we may be approaching the limits of practical falsifiability with particle physics experiments. There could be theories which are logically falsifiable but require probing the extreme energies of the early Universe that we have no way of reproducing. If they are not practically falsifiable, do the theories still count as science?

Another issue is whether falsifiability is necessary or sufficient. I think many would agree that it is an important criterion for a scientific theory, but other things are falsifiable that we would not think of as science. For example, astrology makes claims that can be proven false, so does that mean it counts as science?

Puzzle-solving and Paradigms: Kuhn took a different approach to the demarcation problem. Whereas Popper's falsificationism is normative, describing what science should be, Kuhn on the other hand takes a descriptive approach of what science is in practice. He says that most of science (normal science) attempts to solve puzzles within accepted paradigms, and only rarely does "revolutionary science" set about falsifying theories through paradigm shifts when there is great tension within the accepted paradigm. [At some point I asked the audience who thought they had ever falsified a theory. The one or two confident affirmatives is rather small if falsification is the business of science.]

Scientific Underdetermination: Both Popper and Kuhn attempt to describe what science is, or should be, but can science actually do (or know) what it claims to? A compelling account of the epistemology of science (Duhem), and of knowledge in general (Quine), is that our beliefs are underdetermined by the evidence. The evidence by itself is simply not sufficient to determine our scientific beliefs.

One of the ways this could happen is called holistic underdetermination: experiments don't just test focal theories but rather a web of interconnected beliefs, including assumptions about the instrument functionality, data analysis, corollary theories, etc. So we may want to test (or falsify) a given theory, but there are many contributing factors besides that theory that need to be taken into account. If you find a disagreement between theory and observation, you don't automatically know which one of these factors is at fault.

A good example is the apparent discovery of faster-than-light neutrinos. The experiment could have falsified special relativity, and indeed there was a lot of media hype. But most scientists were unwilling to let special relativity go so easily, and eventually a loose cable was found to be the culprit. Science didn't tell us whether the correct interpretation of the data was falsifying special relativity or finding a loose cable, rather it was our intuition (and unknown philosophical assumptions?) that led us to search very hard for another explanation. This example is a bit rare, however; most of the time our results aren't in danger of falsifying e.g. special relativity, so while there could always be a bug in our code, we aren't always motivated to go looking for it. (And let's face it, there is most likely a bug in our code.)

Another type of underdetermination is called contrastive: evidence that supports one theory may also support an alternative, whether this is known or not. A good example of this is the past history of science: when we were making observations that supported Newtonian gravity, we did not know that they also supported General Relativity or that GR is more accurate than Newtonian gravity. It is likely that our current theories will change in the future, but the current evidence underdetermines the choice between our current best theory and some unknown future theory.

Scientific Explanation

Now we get to the question: what counts as scientific explanation? This is one of the more important philosophical questions for scientists to think about, in my opinion, but it turns out there are a lot of  thorny difficulties in developing a coherent theory of scientific explanation.

The Deductive-Nomological model might be the most well-known, and at this point I will have to throw some Latin at you. You can't have philosophy without Latin. The terminology is meant to make the language easier, so rather than saying "the thing doing the explaining" over and over again you say explanans, and "the phenomenon being explained" is the explanandum. The two components of the DN model are:

1. The explanandum must logically follow from the explanans, and
2. The explanans must contain at least one "law of nature" (as yet undefined).  

The first part is the "deductive" and the second is the "nomological" i.e. dealing with physical law.

As with most things in philosophy, this seems perfectly reasonable on its face but has several problems. The first deals with explanatory asymmetries. Suppose you want to explain the length, s, of a shadow cast by a pole. Then s is the explanandum, and it is explained by the height of the pole, h,  the angle of the Sun, A, and a law describing the propagation of light [and some trigonometry...]. But, you could also take h as the explanandum and explain it the same way given the length of the shadow, and it would fit within the deductive-nomological model. But something about that doesn't seem right - it makes sense for the height of the pole to explain the length of the shadow, but not for the length of the shadow to explain the height of the pole. The issue is probably wrapped up in the notion of causality, which the model fails to address. (We have some notion of how the shadow is caused by the pole and not the other way around, so the explanation should only work in one direction.)

Another problem is that explanatory irrelevancies seem to count as explanation when they shouldn't. Nothing in the DN model distinguishes the relevant from the irrelevant pieces of explanation. The irrelevant bit is usually in the law (or generalization) part of the explanans, for example, "all males who take birth control pills fail to get pregnant;" the fact that they take birth control is irrelevant to explaining why they fail to get pregnant, but the DN model doesn't account for that.

Other models of scientific explanation try to develop an account of what is relevant. According to the Statistical Relevance model, an attribute can only count towards a scientific explanation if it has an effect on the outcome, i.e. the probability that the explanandum occurs. One problem with the SR model is that causal relationships are greatly underdetermined by statistical relevance relationships; many different sets of causal relationships may be compatible with the same SR relationships.

The Causal Mechanical model attempts to account for the role of causation in explanation: an explanation of some event must trace the causal processes and causal interactions (specifically defined) leading up to it and describe the processes and interactions that make up the event. A problem with the CM model is that it provides no way of distinguishing the relevant from irrelevant explanatory features of the causal processes.

There are many other models out there, reflecting how challenging it is to develop a model that describes all the different types of scientific explanations, as well as the nature of physical law and the nature of causality. I personally feel that as scientists, the fact that we don't have an accepted model for what we're doing when we attempt to explain something scientifically should give us pause.

Scientific Realism

Instead of one question in this topic, there are three domains of questions in scientific realism. The first is metaphysical: does nature exist independently of our thoughts and measurements of it? [Most in the audience agreed.] The semantic questions asks: what makes theories true? Are they literal descriptions of nature? [Most in the audience disagreed, though we later found that many interpreted this to mean "exact" descriptions of nature. Clarifying semantic realism as the statement that scientific claims can be interpreted literally (at face value), most then agreed. The irony that talking about semantic realism ran into problems with semantics, however, was lost.] The epistemic question of realism concerns our knowledge of what is real: can we know whether our theories are at least approximately true? [Most agreed.]

It turns out that a poll of professional philosophers found that 75% support a realist position, 12% anti-realist, and 13% other. The poll didn't distinguish between the different domains, however.

Metaphysical realism affirms the mind-independent reality of the world that is investigated by the sciences. This seems obviously true, and maybe it is, but the anti-realist would take care to distinguish this from the "world in itself" and claim that the world investigated by the sciences depends on e.g. theoretical assumptions and perceptual training, so there could be doubts about its mind-independent reality.

Perhaps more interesting is the position of semantic realism that says theories are literal descriptions of nature (i.e. scientific claims can be interpreted literally, at face value). So for example, our theories about electrons, protons, quarks, etc. imply there are physically real things called electrons, protons, and quarks. An anti-realist position (Instrumentalism) would contend that claims about unobservable things have no literal meaning - if all we have are equations about quarks and observations about the effect of quarks, we can't use that to infer that quarks are real unless we can observe them directly.

Finally, the most contentious form of realism is epistemic, which affirms that theoretical claims constitute knowledge about the world. The anti-realist would point out that our past theories have been shown to be false, time and time again, so it is unreasonable for us to believe that our current theories are true; this is called pessimistic meta-induction and has the coolest name ever. In opposition to this, and an argument for realism, the Miracle argument says that realism is the only position that does not make a miracle out of the predictive success of science.

Interlude: Why Care? (Part II)

Now that you know a bit about the philosophy of science, I want to revisit the question of whether scientists should care about philosophy, with some specific examples. First I want to make the practical point that scientists have philosophical assumptions, whether they are aware of them or not, and these affect what they work on and what they fund while on grant panels, so I think it's really worth it for everyone to bring these assumptions out into the open.

The first example concerns what I work on, Modified Gravity (MG), which I took pains to assure people was not the same as MOND when I accepted this postdoc. [For non-cosmologist readers: MG tweaks general relativity to explain the acceleration of the Universe, which is fine because we really don't know what dark energy is; MOND tweaks Newtonian gravity to get rid of dark matter on the scale of galaxies but can't explain all the evidence for dark matter on larger scales and in the early Universe, so most disregard it as a useful theory.] Most modified gravity theories require screening mechanisms to protect the theory from strict Solar System constraints. One might ask, "Why is it necessary to protect the theory at all? Don't Solar System constraints falsify MG?" I think this reflects a lack of understanding of science in practice, and particularly the underdetermination of science: modified gravity is still interesting until we have a successful theory of the accelerated expansion of the Universe, so it is worth it to explore all theoretical avenues including screening mechanisms; others may disagree!

The next example is about Semi-Analytic Models (SAMs), which use both numerical simulations and analytic models to understand how galaxies form and evolve in their dark matter halos. What is the problem with SAMs? ["Too many parameters!" the entire audience was supposed to answer! But at least a couple of people said it...] The common criticism of SAMs is that they have too many parameters; they can fit any observables, so they are in some sense not falsifiable and thus not useful. But is the point of SAMs really to predict all observables, or do they rather explore the relations between the many and varied physical processes of galaxy formation, described by parameters?

As scientists, it's important to be as unbiased as possible, which includes spending some time figuring out what our philosophical assumptions are, what they imply, and whether they are reasonable.

III. Issues in the Philosophy of Cosmology

Finally we get to the Cosmology part of this Philosophy of Cosmology talk [now blog post]. First I want to point people to a really interesting essay written by George Ellis on the philosophy of cosmology, who splits the philosophy of cosmology into several themes:

A. The uniqueness of the Universe: we cannot experiment on the Universe or compare it to others, and concepts of probability are questionable
B. The large scale of the Universe in space and time: observations restricted to our past null cone; limits to determining the geometry of the Universe
C. The unbound energies of the early Universe: we cannot experimentally test extreme energies of the early Universe
D. Explaining the Universe - the question of origins: physics (science) cannot explain the initial state of the Universe, since other specific forms are consistent with physical laws
E. The Universe as the background for existence: physical laws may depend on the nature of the Universe, and cannot be taken for granted
F. The explicit philosophical basis: philosophical choices necessarily inform theory as with all science, but especially because of inherent difficulties re: testability in cosmology
G. The anthropic question and fine tuning for life: life is possible because both the laws of physics and the boundary conditions of the Universe have a very special nature
H. The possible existence of multiverses: unprovable by observation or experimentation, neither can probability-based arguments be used as proof, thus they are philosophical and not scientific proposals
I. Natures of existence: we do not understand the dominant matter components; the existence of infinities is questionable; are the laws of physics prescriptive or descriptive?

 I think it's a good list of the unique questions we face in cosmology, in addition to the usual questions like "What is Science?"

Copernican vs. Cosmological Principles

The Copernican principle is a philosophical principle which claims that we do not live in a special place in the Universe - there are no privileged observers. It's named after Copernicus, who advocated the heliocentric theory of the Solar System which put the Sun at the center instead of the Earth. This is a philosophical principle because you cannot test it, you can only argue against it. Indeed, it is hard to think of what "special place" even means scientifically. It is also important to note that throughout most of human history, we firmly believed that we WERE in a special place.

A scientific principle, on the other hand, can indeed be tested. The cosmological principle states that the Universe is statistically homogeneous and isotropic, meaning it is homogeneous (looks about the same in all locations) and isotropic (looks the same in all directions) when averaged over large enough scales. It is still a bit unclear what those scales are, but this principle is generally taken for granted, and indeed it is fundamental to modern cosmology.

For some reason, these two are mixed up all the time. I have a list that I add to whenever I come across an abstract that gets it wrong or is problematic. Most of the time, people say things like "we are going to test the Copernican principle" when they really mean the cosmological principle, which is annoying but a minor error. Sometimes though they really mean they are going to test the Copernican principle that we are not in a special place in the Universe, which is worse. Perhaps these offenders are interpreting "special place" to mean statistically rare, which I think misses the point of the Copernican principle. The center of the Universe, if such a place existed (and we don't think it does), would be singularly and uniquely special, for example, and a theory that put the Earth in such a place would smack of supernaturalism. [The acceptance of Naturalism as a foundation of modern science is an interesting tangent, but one that I did not bring up during my talk, so we'll leave it for another day.]

The cosmological principle, because it is scientific, can and has been tested. The isotropy of the Universe is very strongly measured by the cosmic microwave background (CMB): the radiation reaching us from the early Universe looks the same in all directions to about one part in 100000. Homogeneity, however, is more problematic - it is in principle testable, but so far the best argument relies on the Copernican principle:

1. We observe isotropy, especially in the CMB.
2. We use the Copernican principle to assume that since we are not privileged observers, isotropy exists elsewhere (and possibly everywhere).
3. Isotropy everywhere implies a homogeneous Universe, mathematically.

The cosmological principle is fundamental for the use of the FLRW metric in cosmology, which allows us to calculate cosmological space-time distances given some parameters that describe the constituents of the Universe. Other metrics are possible, for example one describing a spherically symmetric space-time around a black hole. However, the cosmological principle doesn't clarify on what scales the assumption of isotropy and homogeneity is valid. Clearly the Universe isn't homogeneous or isotropic on the scale of people, or even the solar system or the local group of galaxies. But isotropy and homogeneity have precise meanings, and the scale of homogeneity can potentially be measured with future experiments - unlike the idea of "specialness" in the Copernican principle.

Another point is that even the above argument doesn't guarantee isotropy (and thus homogeneity) at much earlier times than decoupling or much later times than today, and it doesn't apply far outside the visual horizon because we have no information on the isotropy of the CMB there. So while the cosmological principle is a justifiable assumption that is fundamental to modern cosmology, it is important to note that it is just an assumption.

The Anthropic Principle: Fine Tuning and Multiverses

The anthropic principle, or at least its cosmological version, essentially addresses whether the Universe itself is a special place. The observation that concerns people is that the laws of physics and chemistry, and the values of the fundamental constants, are such that life is possible, and small deviations would make the Universe inhospitable to life. This apparent fine-tuning for life seems to require explanation - why is it that the Universe is such as to create us? [I made the following note to myself in my... notes... but didn't bring it up during the talk, but I think it's worth including here: the laws of physics and values of the constants require explanation, but the fact that they are such that life is possible seems secondary and irrelevant to me (in contradiction to Ellis, but possibly also supported by Smolin). See more links below.]

The Weak Anthropic Principle points out that the Universe must be compatible with the existence of us as observers because of the very fact that we exist. It's practically a tautology, but at least it reduces the apparent fine tuning to a selection bias. However, it doesn't attempt to explain the nature of the laws or values of the constants that lead to the apparent fine tuning.

The Strong Anthropic Principle, on the other hand, asserts that our existence as intelligent observers is an explanation for the nature of the laws of physics and chemistry and the values of the constants - the Universe is hospitable to life because we exist.

The explanatory nature of the strong anthropic principle only really makes sense in the context of the multiverse: given the possibility of many different universes with different parameters, of course we exist in the one which is capable of producing us to observe it.

The anthropic principle by itself wasn't regarded as a way to address the apparent fine tuning of the Universe for life - until theories suggested the possibility of the multiverse. For example, the chaotic inflation of Guth proposes that there are many disconnected expanding universe domains. Also, the landscape of string theory posits a huge number (10^500 - that's a 1 with 500 zeros after it) of string vacua in which the effective physical constants could be different. [Seems like a very large case of scientific underdetermination to me...]

A severe problem with positing the existence of multiple universes in order to explain the fine tuning problem is that multiverses are inherently untestable, since they are by definition causally disconnected from our universe (or domain).

Another problem is that another set of cosmological parameters has been shown to be hospitable to life (producing galaxies, chemistry, and long-lived stars), yet the parameters in this Cold Big Bang model are ruled out by the data. An anthropic argument meant to explain the parameters of our Universe based on it being fine-tuned for life would have to explain why we don't live in a Cold Big Bang universe, and they do not. For me, this puts the nail in the coffin of anthropic arguments masquerading as science.

As a final note (and not included in my talk), some of my issues with discussions of the "multiverse" come down to semantics. Sure, you can joke that all philosophers do is argue semantics, but it turns out that the meanings of our words are important if we want to think deeply and communicate our thoughts to others. So, if you want to talk about how chaotic inflation predicts many disconnected expanding universe domains, I'm not comfortable with calling that a "multiverse". A multiverse means many universes, not many domains within our universe - it is not clear to me that the expanding domains are intrinsically causally disconnected, at all times and places (i.e. what about the boundaries?). Whereas calling something a "multiverse" is more in line with the many worlds interpretation of quantum mechanics: we define the Universe as the ultimate set of all that there is in space-time, but we can imagine different ultimate sets, and the set of all ultimate sets we call the multiverse. Without epistemic access to this multiverse, it is difficult to see how multiverse theories can be scientific - but as we saw earlier, what science is and how it does its explaining are still debated in philosophy.


Further Reading

For further reading on philosophy in general, the philosophy of cosmology, etc. check out:

• The Stanford Encyclopedia of Philosophy
• George Ellis, "Issues in the Philosophy of Cosmology"
• Scientia Salon blog (which unfortunately ended over the Summer)
• Rationally Speaking blog (archives)
• Preposterous Universe blog (Sean Carroll)

And for more on the Anthropic Principle:

• Steven Weinberg, "Anthropic bound on the cosmological constant"
• Lee Smolin, "Scientific alternatives to the anthropic principle"
• George Ellis, Note on "Large number coincidences and the anthropic principle in cosmology"
• Anthony Aguirre, "The Cold Big Bang as a counter-example to several anthropic arguments"