The knower paradox states that the statement ‘We know that this statement is false’ leads to inconsistency. This article presents a fresh look at this paradox and some well-known solutions from the literature. Paul Égré discusses three possible solutions that modal provability logic provides for the paradox by surveying and comparing three different provability interpretations of modality, originally described by Skyrms, Anderson, and Solovay. In this article, some background is explained to clarify Égré’s solutions, all three of which hinge (...) on intricacies of provability logic and its arithmetical interpretations. To check whether Égré’s solutions are satisfactory, we use the criteria for solutions to paradoxes defined by Susan Haack and we propose some refinements of them. This article aims to describe to what extent the knower paradox can be solved using provability logic and to what extent the solutions proposed in the literature satisfy Haack’s criteria. Finally, the article offers some reflections on the relation between knowledge, proof, and provability, as inspired by the knower paradox and its solutions. (shrink)

Stereotypes shape inferences in philosophical thought, political discourse, and everyday life. These inferences are routinely made when thinkers engage in language comprehension or production: We make them whenever we hear, read, or formulate stories, reports, philosophical case-descriptions, or premises of arguments – on virtually any topic. These inferences are largely automatic: largely unconscious, non-intentional, and effortless. Accordingly, they shape our thought in ways we can properly understand only by complementing traditional forms of philosophical analysis with experimental methods from psycholinguistics. This (...) paper seeks, first, to bring out the wider philosophical relevance of stereotypical inference, well beyond familiar topics like gender and race. Second, we wish to provide philosophers with a toolkit to experimentally study these ubiquitous inferences and what intuitions they may generate. This paper explains what stereotypes are, and why they matter to current and traditional concerns in philosophy – experimental, analytic, and applied. It then assembles a psycholinguistic toolkit and demonstrates through two studies how potentially questionnaire-based measures can be combined with process measures to garner evidence for specific stereotypical inferences and study when they ‘go through’ and influence our thinking. (shrink)

In this paper I shall defend the idea that there is an abstract and general core meaning of objectivity, and what is seen as a variety of concepts or conceptions of objectivity are in fact criteria of, or means to achieve, objectivity. I shall then discuss the ideal of value-free science and its relation to the objectivity of science; its status can be at best a criterion of, or means for, objectivity. Given this analysis, we can then turn to the (...) problem of inductive risk. Do the value judgements regarding inductive risk really pose a threat to the objectivity of science? I claim that this is not the case because they do not lower the thresholds scientifically postulated for objectivity. I shall conclude the paper with a discussion of under-appreciated influences of values on science, which indeed pose a serious threat to the objectivity of some scientific disciplines. (shrink)

The ideas behind open source software are currently applied to the production of encyclopedias. A sample of six English text-based, neutral-point-of-view, online encyclopedias of the kind are identified: h2g2, Wikipedia, Scholarpedia, Encyclopedia of Earth, Citizendium and Knol. How do these projects deal with the problem of trusting their participants to behave as competent and loyal encyclopedists? Editorial policies for soliciting and processing content are shown to range from high discretion to low discretion; that is, from granting unlimited trust to limited (...) trust. Their conceptions of the proper role for experts are also explored and it is argued that to a great extent they determine editorial policies. Subsequently, internal discussions about quality guarantee at Wikipedia are rendered. All indications are that review and ?super-review? of new edits will become policy, to be performed by Wikipedians with a better reputation. Finally, while for encyclopedias the issue of organizational trust largely coincides with epistemological trust, a link is made with theories about the acceptance of testimony. It is argued that both non-reductionist views (the ?acceptance principle? and the ?assurance view?) and reductionist ones (an appeal to background conditions, and a?newly defined??expertise view?) have been implemented in editorial strategies over the past decade. (shrink)

I will first discuss a peculiarity of the realism-antirealism debate. Some authors defending antirealist positions in a philosophical discussion seem to be inconsistent with what they do when treating scientific subjects. In the latter situation, they behave as realists. This tension can be dissolved by distinguishing different discourses belonging to different levels of philosophical radicality. Depending on the respective level, certain presuppositions are either granted or questioned. I will then turn to a discussion of the miracle argument by discussing a (...) simple example of curve fitting. In the example, multiple use-novel predictions are possible without indicating the truth of the fitting curve. Because this situation has similarities with real scientific cases, it sheds serious doubt upon the miracle argument. Next, I discuss the strategy of selective realism, especially its additional crucial component, the continuity argument. The continuity of some X in a series of theories, with X being responsible for the theories’ use-novel predictions, is taken to be a reliable indicator for the reality of X. However, the continuity of X could as well be due to the similarity of the theories in the series with an empirically very successful theory embodying X, without X being real. Thus, the two main arguments for scientific realism show severe weaknesses. (shrink)

It is still a live question in epistemology and philosophy of science as to what exactly evidence is. In my view, evidence consists in experiences called “seemings.” This view is a version of the phenomenal conception of evidence, the position that evidence consists in nonfactive mental states with propositional content. This conception is opposed by sense-data theorists, disjunctivists, and those who think evidence consists in physical objects or publicly observable states of affairs—call it the courtroom conception of evidence. Thomas Kelly (...) has recently argued that the phenomenal conception cannot play all the roles evidence plays and is thus inadequate. Having first explained the nature of seemings, in this essay I utilize Kelly’s own understanding of the four major roles of evidence and argue that the phenomenal conception can play each one. Experience is a good candidate for evidence. (shrink)

Van Inwagen proposes that besides simples only living organisms exist as composite objects. This paper suggests expanding van Inwagen’s ontology by also accepting composite objects in the case that physical bonding occurs (plus some extra conditions). Such objects are not living organ-isms but rather physical bodies. They include (approximately) the complete realm of inanimate ordinary objects, like rocks and tables, as well as inanimate scientific objects, like atoms and mol-ecules, the latter filling the ontological gap between simples and organisms in (...) van Inwagen’s origi-nal picture. We thus propose a compositional pluralism claiming that composition arises if and on-ly if bonding or life occurs. (shrink)

The existence of God is once again the focus of vivid philosophical discussion. From the point of view of analytic theology, however, people often talk past each other when they debate about the putative existence or nonexistence of God. In the worst case, for instance, atheists deny the existence of a God, which no theists ever claimed to exist. In order to avoid confusions like this we need to be clear about the function of the term 'God' in its different (...) contexts of use. In what follows, I distinguish between the functions of 'God' in philosophical contexts on the one hand and in theological contexts on the other in order to provide a schema, which helps to avoid confusion in the debate on the existence or non-existence of God. (shrink)

Many virtual communities that rely on user-generated content (such as social news sites, citizen journals, and encyclopedias in particular) offer unrestricted and immediate ‘write access’ to every contributor. It is argued that these communities do not just assume that the trust granted by that policy is well-placed; they have developed extensive mechanisms that underpin the trust involved (‘backgrounding’). These target contributors (stipulating legal terms of use and developing etiquette, both underscored by sanctions) as well as the contents contributed by them (...) (patrolling for illegal and/or vandalist content, variously performed by humans and bots; voting schemes). Backgrounding trust is argued to be important since it facilitates the avoidance of bureaucratic measures that may easily cause unrest among community members and chase them away. (shrink)

The role of certitude in our mental lives is, to put it mildly, controversial. Many current epistemologists (including epistemologists of religion) eschew certitude altogether. Given his emphasis on certitude, some have maintained that John Henry Newman was an infallibilist about knowledge. In this paper, we argue that a careful examination of his thought (especially as seen in the Grammar of Assent) reveals that he was an epistemic fallibilist. We first clarify what we mean by fallibilism and infallibilism. Second, we explain (...) why some have read Newman as an infallibilist. Third, we offer two arguments that Newman is at least a fallibilist in a weak sense. In particular, the paradox he seeks to resolve in the Grammar and his dispute with John Locke both indicate that he is at least a weak fallibilist. We close with a consideration of whether Newman is a fallibilist in a much stronger sense as well. (shrink)

In this chapter, we argue for a phenomenal conservative perspective on religious epistemology and attempt to answer some common criticisms of this perspective.

From the mid-1960s until the late 1980s, the well-known general philosophies of science of the time were applied to economics. The result was disappointing: none seemed to fit. This paper argues that this is due to a special feature of economics: it possesses ‘orientational paradigms’ in high number. Orientational paradigms are similar to Kuhn’s paradigms in that they are shared across scientific communities, but dissimilar to Kuhn’s paradigms in that they are not generally accepted as valid guidelines for further research. (...) As will be shown by several examples, orientational paradigms provide economics with common points of reference that support its epistemic coherence and make scientific discourse more easily possible across school boundaries. With the help of systematicity theory, a newer general philosophy of science, one can further elucidate the role of orientational paradigms with regard to scientific progress. (shrink)

Hacker communities of the 1970s and 1980s developed a quite characteristic work ethos. Its norms are explored and shown to be quite similar to those which Robert Merton suggested govern academic life: communism, universalism, disinterestedness, and organized scepticism. In the 1990s the Internet multiplied the scale of these communities, allowing them to create successful software programs like Linux and Apache. After renaming themselves the `open source software' movement, with an emphasis on software quality, they succeeded in gaining corporate interest. As (...) one of the main results, their `open' practices have entered industrial software production. The resulting clash of cultures, between the more academic CUDOS norms and their corporate counterparts, is discussed and assessed. In all, the article shows that software practices are a fascinating seedbed for the genesis of work ethics of various kinds, depending on their societal context. (shrink)

There is a close systematic relationship between panentheism, as a metaphysical theory about the relation between God and the world, and transhumanism, the ethical demand to use the means of the applied sciences to enhance both human nature and the environment. This relationship between panentheism and transhumanism provides a ‘cosmic’ solution to the problem of evil: on panentheistic premises, the history of the world is the one infinite life of God, and we are part of the one infinite divine being. (...) We ourselves are therefore responsible for the future development of the life of the divine being. We should therefore use the means provided by the natural sciences to develop the history of the world in such a way that the existence of evil shall be overcome and shall no longer be part of the divine being in whom we move and live and have our being. The metaphysics of panentheism leads to the ethics of transhumanism. (shrink)

A discussion of several tensions between Thomistic philosophy and modern Darwinian theory as well as several recent Thomistic criticisms of intelligent design.

Recent authors, emphasizing Newman’s distaste for natural theology—especially William Paley’s design argument—have urged us to follow Newman’s lead and reject design arguments. But I argue that Newman’s own argument for God’s existence (his argument from conscience) fails without a supplementary design argument or similar reason to think our faculties are truth-oriented. In other words, Newman appears to need the kind of argument he explicitly rejects. Finding Newman’s rejection of natural theology to stem primarily from factors other than worries about cogency, (...) however, I further argue that there is little reason not to pursue design arguments in order to save the argument from conscience. (shrink)

In order to fight massive vandalism the English- language Wikipedia has developed a system of surveillance which is carried out by humans and bots, supported by various tools. Central to the selection of edits for inspection is the process of using filters or profiles. Can this profiling be justified? On the basis of a careful reading of Frederick Schauer’s books about rules in general (1991) and profiling in particular (2003) I arrive at several conclusions. The effectiveness, efficiency, and risk-aversion of (...) edit selection all greatly increase as a result. The argument for increasing predictability suggests making all details of profiling manifestly public. Also, a wider distribution of the more sophisticated anti-vandalism tools seems indicated. As to the specific dimensions used in profiling, several critical remarks are developed. When patrollers use ‘assisted edit- ing’ tools, severe ‘overuse’ of several features (anonymity, warned before) is a definite possibility, undermining profile efficacy. The easy remedy suggested is to render all of them invisible on the interfaces as displayed to patrollers. Finally, concerning not only assisted editing tools but tools against vandalism generally, it is argued that the anonymity feature is a sensitive category: anons have been in dispute for a long time (while being more prone to vandalism). Targeting them as a special category violates the social contract upon which Wikipedia is based. The feature is therefore a candidate for mandatory ‘underuse’: it should be banned from all anti-vandalism filters and profiling algorithms, and no longer be visible as a special edit trait. (shrink)

Karl Christian Friedrich Krause left an impressive oeuvre consisting of 256 books and articles, covering numerous branches of philosophy, the humanities, and science.[1] His Urbild der Menschheit, his Vorlesungen über das System der Philosophie and his Vorlesungen über die Grundwahrheiten der Wissenschaft are of particular pertinence for philosophers today. [1] See: E. M. Ureña and E. Fuchs, “Einführung in das Gesamtwerk”, in Karl Christian Friedrich Krause. Band 1: Entwurf des Systems der Philosophie, ed. T. Bach and O. Breidbach.

Special divine action is an integral part of the Christian worldview. In fact, the plausibility of the Christian worldview depends on and is grounded in the putative reality, and therefore possibility, of special divine action. Without special divine action, Scripture does not make sense, and without Scripture, Christianity neither. However, the possibility of special divine action is highly contested in almost every field of human enquiry. In what follows, I briefly suggest a minimal definition of special divine action and show (...) its indispensability for the internal plausibility of Christian faith. I then argue against the very possibility of special divine action. I end by way of identifying ways in which Christian theologians can respond to the arguments in order to justify the possibility of special divine action. It turns out that special divine action neither contradicts science nor metaphysics. (shrink)

In this paper, I shall defend two main claims. First, Friedman’s famous paper “On the methodology of positive economics” (“F53”) cannot be properly understood without taking into account the influence of three authors who are neither cited nor mentioned in the paper: Max Weber, Frank Knight, and Karl Popper. I shall trace both their substantive influence on F53 and the historical route by which this influence took place. Once one has understood these ingredients, especially Weber’s ideal types, many of F53’s (...) astonishing sentences like “the more significant the theory, the more unrealistic the assumptions”, make good sense. Second, I shall claim that the much-discussed question whether Friedman’s essay espouses an instrumentalist or a realist position, is the wrong question to be asked. I shall illustrate that by a comparison with examples from physics in which also unrealistic assumptions are made. Also there, the question whether these assumptions are indicators of instrumentalism or realism is not appropriate. Cleared from these misunderstandings, F53 presents itself as an interesting and reasonable but much less controversial contribution to the methodology of economics. (shrink)

Darcy's law is a phenomenological relationship for fluid flow rate that finds one of its principle applications in hydrology. Theoretical hydrologists rely upon a multiplicity of conceptual models to carry out approximate derivations of Darcy's law. These derivations provide structural explanations of the law; they require the application of fundamental principles, such as conservation of momentum, to idealized models of the porous media within which the flow occurs. In practice, recognition of the idealized conditions incorporated into models facilitates the empirical (...) clarification of the domain within which the law remains accurate. Structural explanations also contribute to the physical interpretation of phenomenological parameters. (shrink)

In a time when conservatives believe that the traditional family is under increasing fire, some think an appeal to Darwinian science may be the answer. I argue that these conservatives are wrong to maintain that Darwinian theory can serve as the intellectual foundation for the traditional conception of the family. Contra Larry Arnhart and James Q. Wilson, a Darwinian philosophy of nature simply lacks the stability the traditional family requires; it cannot support the traditional conception of human nature and the (...) normativity it was thought to contain. If conservatives are to maintain these traditional ideas, the theoretical foundation must lie elsewhere. (shrink)

Physics tells us that the world that we live in is “really” composed of vast numbers of tiny particles, defined and individuated by their mass, charge, velocity and position. That is all there is, the rest being empty space. In contrast the world that we interact with in our every day lives is composed of rooms with windows, people and cats, and plates filled with milk and cornflakes. It is a world of individual objects and stuffs, most of which can (...) be effortlessly identified and labelled as being of a particular kind. Like an illustration in a child’s reader, we could look around our environment and attach labels to all of the objects and kinds of things around us. (shrink)

(VietBuzz) – i2Metrix – bộ chỉ tiêu đo lường năng lực đổi mới sáng tạo (ĐMST) doanh nghiệp do BSA và DHVP Research phát triển – được thực hiện trong hai tháng đầu năm 2014 với 19 thành viên của CLB Doanh nghiệp Dẫn đầu (BSA/LBC) trước tiên ghi nhận đóng góp trách nhiệm xã hội của nhóm doanh nghiệp sớm nhận thức giá trị của ĐMST toàn diện trong sản xuất và kinh doanh.

(Mặt trận) - Chế độ sở hữu trong nền kinh tế thị trường định hướng xã hội chủ nghĩa Việt Nam trước hết phải tuân theo các nguyên tắc của nền kinh tế thị trường hiện đại. Trong các nguyên tắc của nền kinh tế thị trường hiện đại, nguyên tắc sở hữu tư nhân là nền tảng của nền kinh tế thị trường - là nguyên tắc quan trọng. Xa rời nguyên tắc này, dù chúng ta cố gắng xây (...) dựng nền kinh tế thị trường thì đó vẫn là một nền kinh tế thị trường khó được quốc tế công nhận. Sở hữu tư nhân hiện đại là chế độ sở hữu được cổ phần hóa, được xã hội hóa rất cao, công khai, minh bạch trên các thị trường chứng khoán. Tính xã hội cao của chế độ cổ phần hóa đã được C.Mác đánh giá, xem như là chế độ sở hữu quá độ sang hình thái kinh tế - xã hội mới. (shrink)

Nghiên cứu được thực hiện nhằm xác định các yếu tố ảnh hưởng đến Ý định đặt phòng khách sạn trực tuyến của giới trẻ tại Đà Nẵng, dựa trên dữ liệu khảo sát bằng bảng hỏi cấu trúc. Kết quả nghiên cứu cho thấy, Ý định đặt phòng của giới trẻ tại Đà Nẵng chịu tác động của 3 yếu tố là: (1) Chất lượng trang web; (2) Sự gắn kết; (3) Tương tác xã hội. Từ kết quả nghiên (...) cứu, tác giả đã đưa ra những khuyến nghị nhằm khuyến khích giới trẻ sử dụng phương pháp trực tuyến để đặt phòng khi đi du lịch từ đó kích thích ngành du lịch ngày càng phát triển hơn ở Đà Nẵng. (shrink)

Nghiên cứu phân tích các yếu tố ảnh hưởng đến mức độ hài lòng của khách hàng khi sử dụng dịch vụ truyền hình số mặt đất trên địa bàn tỉnh Vĩnh Long thông qua việc thu thập dữ liệu từ 150 khách hàng. Kết quả nghiên cứu cho thấy, có 6 yếu tố chính tác động đến sự hài lòng của khách hàng, bao gồm: Phong cách phục vụ, Chất lượng nội dung, Chất lượng tín hiệu, Cơ sở hạ (...) tầng, Đội ngũ kỹ thuật và Độ uy tín của công ty. Từ kết quả thu được, nghiên cứu đề xuất một số giải pháp nhằm giúp các doanh nghiệp trong ngành gia tăng sự hài lòng của khách hàng. (shrink)

A review of Edward Feser's Five Proofs of the Existence of God.

What was René Girard’s attitude towards philosophy? What philosophers influenced him? What stance did he take in the philosophical debates of his time? What are the philosophical questions raised by René Girard’s anthropology? In this interview, Paul Dumouchel sheds light on these issues.

In the replies to my critics that follow I offer a more detailed account of the specific papers that they discuss or examine. The papers that they are especially concerned with are: “The Material World and Natural Religion in Hume’s Treatise” (Ryan) [Essay 3], “Hume’s Skepticism and the Problem of Atheism” (Fosl) [Essay 12], and “Hume’s Philosophy of Irreligion and the Myth of British Empiricism (Gautier) [Essay 16].

Recasting Hume and Early Modern Philosophy is a collection of essays that are all concerned with major figures and topics in the early modern philosophy. Most of the essays are concerned, more specifically, with the philosophy of David Hume (1711-1776). The sixteen essays included in this collection are divided into five parts. These parts are arranged under the headings of: (1) Metaphysics and Epistemology; (2) Free Will and Moral Luck; (3) Ethics, Virtue and Optimism; (4) Skepticism, Religion and Atheism; and (...) (5) Irreligion and the Unity of Hume’s Thought. A particularly important theme running through many of these essays is the subject of Hume’s irreligious aims and intentions. The fifth and final part of the collection is devoted to an articulation and defence of this specific understanding of Hume’s philosophical thought. (shrink)

The Limits of Free Will presents influential articles by Paul Russell concerning free will and moral responsibility. The problems arising in this field of philosophy, which are deeply rooted in the history of the subject, are also intimately related to a wide range of other fields, such as law and criminology, moral psychology, theology, and, more recently, neuroscience. These articles were written and published over a period of three decades, although most have appeared in the past decade. Among the (...) topics covered: the challenge of skepticism; moral sentiment and moral capacity; necessity and the metaphysics of causation; practical reason; free will and art; fatalism and the limits of agency; moral luck, and our metaphysical attitudes of optimism and pessimism. (shrink)

For Rawls, the demands of justice compete with moral and religious obligations that are part of citizens’ comprehensive doctrines. The ways we love are shaped by our comprehensive doctrines; however, love can also stand in opposition to our moral and religious beliefs. I will argue that love – spousal, familial and associational – constitutes its own register of values along with its own set of obligations. For this reason love confronts not only our moral and religious beliefs, it also confronts (...) the demands and requirements of justice. Rawls has worked out a negotiated peace between the obligations of justice and the demands of citizens’ comprehensive doctrines in the form of an overlapping consensus on a free-standing political conception of justice. I will argue that in Rawls’s philosophy, justice has not yet made peace with love. (shrink)

Suppose we accept Nietzsche’s claim that critical reflection undermines our evaluative commitments. Then it seems that we are left with a pair of unappealing options: either we engage in critical reflection and find our evaluative commitments becoming etiolated; or we somehow immunize certain evaluative commitments from the effects of critical reflection. Nietzsche considers both of these paths, labeling the person who results from the first path “the last man” and the person who results from the second “the fanatic.” I consider (...) Nietzsche’s analysis of these two character types, discuss why he thinks that in modernity these are the options with which we are faced, and ask whether Nietzsche thinks that there is a third way. (shrink)

Among recent objections to Pascal's Wager, two are especially compelling. The first is that decision theory, and specifically the requirement of maximizing expected utility, is incompatible with infinite utility values. The second is that even if infinite utility values are admitted, the argument of the Wager is invalid provided that we allow mixed strategies. Furthermore, Hájek has shown that reformulations of Pascal's Wager that address these criticisms inevitably lead to arguments that are philosophically unsatisfying and historically unfaithful. Both the objections (...) and Hájek's philosophical worries disappear, however, if we represent our preferences using relative utilities rather than a one-place utility function. Relative utilities provide a conservative way to make sense of infinite value that preserves the familiar equation of rationality with the maximization of expected utility. They also provide a means of investigating a broader class of problems related to the Wager. (shrink)

The idea of epistemic temperance is introduced and explicated through a discussion of Plato's understanding of it. A variety of psychological and epistemic phenomena are presented which arise due to epistemic intemperance, or the inappropriate influence of conations on cognition. Two cases familiar to philosophers, self-deception and racial prejudice, are discussed as the result of epistemic intemperance though they are not typically seen as having a common cause. Finally, epistemic temperance is distinguished from epistemic justice, as these have been conflated.

THE PRINCIPLE OF SUPERPOSITION. The need for a quantum theory Classical mechanics has been developed continuously from the time of Newton and applied to an ...

De Finetti would claim that we can make sense of a draw in which each positive integer has equal probability of winning. This requires a uniform probability distribution over the natural numbers, violating countable additivity. Countable additivity thus appears not to be a fundamental constraint on subjective probability. It does, however, seem mandated by Dutch Book arguments similar to those that support the other axioms of the probability calculus as compulsory for subjective interpretations. These two lines of reasoning can be (...) reconciled through a slight generalization of the Dutch Book framework. Countable additivity may indeed be abandoned for de Finetti's lottery, but this poses no serious threat to its adoption in most applications of subjective probability. Introduction The de Finetti lottery Two objections to equiprobability 3.1 The ‘No random mechanism’ argument 3.2 The Dutch Book argument Equiprobability and relative betting quotients The re-labelling paradox 5.1 The paradox 5.2 Resolution: from symmetry to relative probability Beyond the de Finetti lottery. (shrink)

In the 6th century, Paul the Persian used his own pen to write a summary of Aristotle's Peri Hermeneias in the Persian language. Severus Sebokht translated it into Syriac. This book is a transcription and translation of the Syriac manuscript of Paul the Persian's Peri Hermeneias and a comparison of it with Aristotle's original Greek text.

According to Augustine, abstract objects are ideas in the mind of God. Because numbers are a type of abstract object, it would follow that numbers are ideas in the mind of God. Call such a view the “Augustinian View of Numbers” (AVN). In this paper, I present a formal theory for AVN. The theory stems from the symmetry conception of God as it appears in Studtmann (2021). I show that the theory in Studtmann’s paper can interpret the axioms of Peano (...) Arithmetic minus the induction schema. This fact allows for the development of arithmetic in a natural way. The development eventuates in a theory that can interpret second-order arithmetic. The conception of God that emerges by the end of the discussion is a conception of an infinite, ineffable, self-cause that contains objects that not only serve as numbers but also encode information about each other. (shrink)