Collatz conjecture - In reverse. There is another approach to prove the following conjecture, which considers the bottom-up method of growing the Collatz graph. The Collatz graph is defined by an inverse relation, So, instead of proving that all natural numbers eventually lead to 1, we can prove that 1 leads to all natural numbers. Also, the inverse relation forms a tree except for the 1-2 loop. Note that the relation being inverted here is (3n + 1) / 2 (see Optimizations below). ...

See also:

Collatz conjecture, Collatz conjecture - Statement of the problem, Collatz conjecture - Examples, Collatz conjecture - Program to calculate Collatz sequences, Collatz conjecture - Supporting arguments, Collatz conjecture - Experimental evidence, Collatz conjecture - Probabilistic evidence, Collatz conjecture - Other ways of looking at it, Collatz conjecture - In reverse, Collatz conjecture - As rational numbers, Collatz conjecture - As an abstract machine, Collatz conjecture - As iterating a real or complex map, Collatz conjecture - Optimizations

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Bundle theory is the ontological theory about objecthood in which an object consists only of a collection (bundle) of properties. According to bundle theory, an object consists of its properties and nothing more, thus neither can there be an object without properties nor can one even conceive of such an object. For example, bundle theory claims that thinking of an apple compels one also to think of its color, its shape, the fact that it is a kind of fruit, its cells, its taste, or at least one other of its propert ...

Including:

• Bundle theory - Arguments for the bundle theory
• Bundle theory - Objections to the bundle theory
• Bundle theory - Compressence objection
• Bundle theory - Language-reality objection
• Bundle theory - Bundle Theory and Eastern Philosophy

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In mathematics, a binary relation, sometimes called dyadic relation, is a relation between two entities. It is exemplified by such ideas as "is greater than" or "is equal to" in arithmetic, "is congruent to" in geometry, and "is an element of" or "is a subset of" in set theory. Functions are also a special case of binary relations (the difference between a function and a binary relation is that a function must have a unique output for every input while relations may have any number of outputs for one input). Put in lay t ...

Including:

• Binary relation - Definition and examples
• Binary relation - Definition
• Binary relation - Remark
• Binary relation - Example
• Binary relation - Special types of relations
• Binary relation - Relations over a set
• Binary relation - Operations on binary relations
• Binary relation - Examples of common binary relations

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Analogy is either the cognitive process of transferring information from a particular subject (the analogue or source) to another particular subject (the target), or a linguistic expression corresponding to such a process. In a narrower sense, analogy is an inference or an argument from a particular to another particular, as opposed to deduction, induction and abduction, where at least one of the premises or the conclusion is general. The word analogy can also refer to the relation between the source and the target themselves, which is often, though not necessarily, a simil ...

Including:

• Analogy - Models and theories of analogy
• Analogy - Identity of relation
• Analogy - Shared abstraction
• Analogy - Special case of induction
• Analogy - Hidden deduction
• Analogy - Shared structure
• Analogy - High-level perception
• Analogy - Applications and types of analogy
• Analogy - Linguistics
• Analogy - Mathematics
• Analogy - Artificial intelligence
• Analogy - Anatomy
• Analogy - Law
• Analogy - Engineering

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An abstraction is an idea, concept, or word which defines the phenomena that make up its referents (those concrete events or things to which the abstraction refers). Abstraction - Thought process. In philosophical terminology abstraction is the thought process wherein ideas are distanced from objects. Abstraction uses a strategy of simplification of detail, wherein formerly concrete details are left ambiguous, vague, or undefined; thus speaking of things in the abstract demands that the listener have ...

Including:

• Abstraction - Thought process
• Abstraction - Conceptual schemes for abstraction
• Abstraction - Referents
• Abstraction - Instantiation
• Abstraction - Physicality
• Abstraction - Realness
• Abstraction - Precise semantic meaning
• Abstraction - Abstraction used in philosophy
• Abstraction - Ontological status
• Abstraction - Reification
• Abstraction - Compression
• Abstraction - The neurology of abstraction
• Abstraction - Abstraction in Art

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In metaphysics (in particular, ontology), the different kinds or ways of being are called categories of being or simply categories. According to the Aristotelian tradition, a being is anything that can be said to be in the various senses of this word. Hence, to investigate the categories of being is to determine the most fundamental senses in which things can be said to be. A category, more precisely, is any of the broadest classes of things - 'thing' here meaning anything whatever that ca ...

Including:

• Category of being - Aristotle's Categories
• Category of being - Categories of being
• Category of being - Physical objects
• Category of being - Minds
• Category of being - Classes
• Category of being - Properties
• Category of being - Relations

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Set theory is a branch of mathematics created principally by the German mathematician Georg Cantor at the end of the 19th century. Initially controversial, set theory has come to play the role of a foundational theory in modern mathematics, in the sense of a theory invoked to justify assumptions made in mathematics concerning the existence of mathematical objects (such as numbers or functions) and their properties. Formal versions of set theory also have a foundational role to play as specifying a theoretical ideal of mathematical rig ...

Including:

• Axiomatic set theory - The origins of rigorous set theory
• Axiomatic set theory - Axioms for set theory
• Axiomatic set theory - Independence in ZFC
• Axiomatic set theory - Set theory ZFC foundations for mathematics
• Axiomatic set theory - Well-foundedness and hypersets
• Axiomatic set theory - Objections to set theory

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Charles Sanders Santiago Peirce (pronounced purse), (September 10, 1839, Cambridge, Massachusetts – April 19, 1914, Milford, Pennsylvania) was an American polymath. Although educated as a chemist and employed as a scientist for 30 years, he is now mostly seen as a philosopher. He is the greatest American builder of architectonic systems, and his admirers deem him the most important systemat ...

Including:

• Charles Peirce - Life
• Charles Peirce - Reception
• Charles Peirce - Works
• Charles Peirce - Peirce's philosophy
• Charles Peirce - Pragmatism
• Charles Peirce - Scholastic realism
• Charles Peirce - Formal perspective
• Charles Peirce - Dynamics of representation
• Charles Peirce - Normative sciences
• Charles Peirce - Parallels with Leibniz
• Charles Peirce - Bibliography
• Charles Peirce - Primary literature
• Charles Peirce - Secondary literature

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In philosophy, the unity of the proposition is the problem of explaining how a sentence in the indicative mood expresses more than just what a list of proper names expresses. Unity of the proposition - History. The problem goes under this name as discussed by Bertrand Russell. But it actually goes back to Plato. According to Plato's dialogue The Sophist, the simplest kind of sentence consists of just a proper name and a universal term (i.e. a predicate). The name refers to or picks out some individua ...

Including:

• Unity of the proposition - History
• Unity of the proposition - Russell Frege Wittgenstein
• Unity of the proposition - Quotation

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In combinatorial mathematics, a cyclic order on a set X with n elements is an arrangement of X as on a clock face, for an n-hour clock. That is, rather than an order relation on X, we define on X just functions 'element immediately before' and 'element immediately following' any given x, in such a way that taking predecessors, or successors, cycles once through the elements as x(1), x(2), ..., x(n). Another way to put it is to say that we make X into the standard n-cycle directed graph on n ...

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The correspondence theory of truth states that something is rendered true by the existence of a fact with corresponding elements and a similar structure. A rejection of any sort of relativism about truth, this theory maintains that the truth or falsity of a statement is determined only by how it relates to the world, and whether it accurately describes (i.e., corresponds with) that world. Problems with the theory arise from consideration of precisely what is supposed to correspond with what. If a statement is just a sentence th ...

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In mathematics, the term well-defined is used to specify that a certain concept (a function, a property, a relation, etc.) is defined in a mathematical or logical way using a set of base axioms in an entirely unambiguous way. One of the most common places in mathematics in which the term well-defined is used is in dealing with cosets in group theory. It is as important that we check that we get the same result regardless of which representative of the coset we choose as it is that we always get the same result when we perform arithmet ...

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In computer programming and some branches of mathematics, strings are sequences of various simple objects. These simple objects are selected from a predetermined set, each entry of which is usually allocated a code. Most commonly these simple objects will be printable characters and the control codes that are used with them. The data types in which these are stored are also called strings and it is fairly common to use these types to store arbitrary, variable-length sequences of binary data. Generally, a string can be placed directly ...

Including:

• String computer science - String datatypes
• String computer science - Representations
• String computer science - Memory management
• String computer science - String algorithms
• String computer science - String oriented languages and utilities
• String computer science - Formal theory

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The Collatz conjecture is an unresolved conjecture in mathematics. It is named after Lothar Collatz, who first proposed it in 1937. The conjecture is also known as the 3n + 1 conjecture, the Ulam conjecture (after Stanislaw Ulam), the Syracuse problem, as the hailstone sequence or hailstone numbers, or as Wondrous numbers as per Gödel, Escher, Bach. It asks whether a certain kind of number sequence always ...

Including:

• Collatz conjecture - Statement of the problem
• Collatz conjecture - Examples
• Collatz conjecture - Program to calculate Collatz sequences
• Collatz conjecture - Supporting arguments
• Collatz conjecture - Experimental evidence
• Collatz conjecture - Probabilistic evidence
• Collatz conjecture - Other ways of looking at it
• Collatz conjecture - In reverse
• Collatz conjecture - As rational numbers
• Collatz conjecture - As an abstract machine
• Collatz conjecture - As iterating a real or complex map
• Collatz conjecture - Optimizations

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In the scientific method, an experiment is a set of actions and observations, performed to support or falsify a hypothesis or research concerning phenomena. The experiment is a cornerstone in the empirical approach to knowledge. See the list of famous experiments for historically important scientific experiments. The word is derived from the Latin ex- + -periri, "from trying". Experiment - An experiment in baking. As a simple example, consider that many bakers have noticed that the amou ...

Including:

• Experiment - An experiment in baking
• Experiment - Design of experiments
• Experiment - Controlled experiments
• Experiment - Natural experiments
• Experiment - Quasi-experiments
• Experiment - Field Experiments
• Experiment - Examples
• Experiment - Quotes
• Experiment - Literature

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Nominalism is the position in metaphysics that there exist no universals outside of the mind. Nominalism is best understood in contrast to realism. Philosophical realism holds that when we use descriptive terms such as "green" or "tree," the Forms of those concepts really exist, independently of world in an abstract realm. Such thought is associated with Plato. Nominalism, by contrast, holds that ideas represented by word ...

Including:

• Nominalism - The problem of universals
• Nominalism - Varieties of nominalism

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Behavior (or behaviour) refers to the actions or reactions of an object or organism, usually in relation to the environment. Behavior can be conscious or unconscious, overt or covert, and voluntary or involuntary. Behavior is controlled by the endocrine system, and the nervous system. The complexity of the behavior of an organism is related to the complexity of its nervous system. Generally, organisms with complex nervous systems have a greater capacity to learn new responses and thus adjust their behavior. of people (and other ...

Including:

• Behavior - External link

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Nominalism arose in reaction to the problem of universals. Specifically, accounting for the fact that some things are of the same type. For example, Fluffy and Kitzler are both cats, or, the fact that certain properties are repeatable, such as: the grass, the shirt, and Kermit the Frog are green. One wants to know in virtue of what makes Fluffy and Kitzler both cats and what makes the grass, the shirt, and Kermit green. The realist answer is that all the green things are green in virtue of the existence of a universal; a ...

See also:

Nominalism, Nominalism - The problem of universals, Nominalism - Varieties of nominalism

Read more here: » Nominalism: Encyclopedia II - Nominalism - The problem of universals

Analogy - Identity of relation. In ancient Greek the word αναλογια (analogia) originally meant proportionality, in the mathematical sense, and it was indeed sometimes translated to Latin as proportio. From there analogy was understood as identity of relation between any two ordered pairs, whether of mathematical nature or not. Kant's Critique of Judgment held to this notion. Kant argued that there can be exactly the same relation between two completely different objects. ...

See also:

Analogy, Analogy - Models and theories of analogy, Analogy - Identity of relation, Analogy - Shared abstraction, Analogy - Special case of induction, Analogy - Hidden deduction, Analogy - Shared structure, Analogy - High-level perception, Analogy - Applications and types of analogy, Analogy - Linguistics, Analogy - Mathematics, Analogy - Artificial intelligence, Analogy - Anatomy, Analogy - Law, Analogy - Engineering

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The problem became significant in the early development of set theory. Set membership is a formal representation of the relation between the two parts of the proposition, and there are certain philosophical problems connected with this, as Frege realised when he investigated the distinction between concept and object. Assume that "Shergar is a horse" analyses into what "Shergar" names (an "Object", according to Frege), and what "is a horse" names (a "Concept"). Objects are fundamentally different from concepts, otherwise we get the problem o ...

See also:

Unity of the proposition, Unity of the proposition - History, Unity of the proposition - Russell Frege Wittgenstein, Unity of the proposition - Quotation

Read more here: » Unity of the proposition: Encyclopedia II - Unity of the proposition - Russell Frege Wittgenstein