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First principle
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First principle
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In philosophy and science, a first principle is a basic proposition or assumption that cannot be deduced from any other proposition or assumption. First principles in philosophy are from first cause[1] attitudes and taught by Aristotelians, and nuanced versions of first principles are referred to as postulates by Kantians.[2]

In mathematics and formal logic, first principles are referred to as axioms or postulates. In physics and other sciences, theoretical work is said to be from first principles, or ab initio, if it starts directly at the level of established science and does not make assumptions such as empirical model and parameter fitting. "First principles thinking" consists of decomposing things down to the fundamental axioms in the given arena, before reasoning up by asking which ones are relevant to the question at hand, then cross referencing conclusions based on chosen axioms and making sure conclusions do not violate any fundamental laws. Physicists include counterintuitive concepts with reiteration.

In formal logic

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In a formal logical system—that is, a set of propositions that are consistent with one another—it is possible that some of the statements can be deduced from other statements. For example, in the syllogism, "All men are mortal; Socrates is a man; Socrates is mortal" the last claim can be deduced from the first two.

A first principle is an axiom that cannot be deduced from any other within that system. The classic example is that of Euclid's Elements; its hundreds of geometric propositions can be deduced from a set of definitions, postulates, and primitive notions: all three types constitute first principles.

Philosophy

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In philosophy, "first principles" are from first cause[1] attitudes commonly referred to as a priori terms and arguments, which are contrasted to a posteriori terms, reasoning, or arguments, in that the former are simply assumed and exist prior to the reasoning process, and the latter are deduced or inferred after the initial reasoning process. First principles are generally treated in the realm of philosophy known as epistemology but are an important factor in any metaphysical speculation.

In philosophy, "first principles" are often somewhat synonymous with a priori, datum, and axiomatic reasoning.

Ancient Greek philosophy

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"Arche" redirects here. For other uses, see Arche (disambiguation).

In ancient Greek philosophy, a first principle from which other principles are derived is called an arche[note 1] and later "first principle" or "element". By extension, it may mean "first place", "method of government", "empire, realm", or "authorities".[note 2] The concept of an arche was adapted from the earliest cosmogonies of Hesiod and Orphism, through the physical theories of Pre-Socratic philosophy and Plato before being formalized as a part of metaphysics by Aristotle. Arche,[note 3] sometimes also transcribed as arkhé, is an Ancient Greek word with primary senses "beginning", "origin" or "source of action",[note 4] from the beginning, οr the original argument, "command".[3] The first principle or element corresponds to the "ultimate underlying substance" and "ultimate indemonstrable principle".[4]

Mythical cosmogonies

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The heritage of Greek mythology already embodied the desire to articulate reality as a whole and this universalizing impulse was fundamental for the first projects of speculative theorizing. It appears that the order of "being" was first imaginatively visualized before it was abstractly thought.[5]

In the mythological cosmogonies of the Near East, the universe is formless and empty and the only existing thing prior to creation was the water abyss. In the Babylonian creation story, Enuma Elish, the primordial world is described as a "watery chaos" from which everything else appeared.[6] This watery chaos has similarities in the cosmogony of the Greek mythographer Pherecydes of Syros.[7] In the mythical Greek cosmogony of Hesiod (8th to 7th century BC), the origin of the world is Chaos, considered as a divine primordial condition, from which everything else appeared. In the creation "chaos" is a gaping-void, but later the word is used to describe the space between the Earth and the sky, after their separation. "Chaos" may mean infinite space, or a formless matter which can be differentiated.[8] The notion of temporal infinity was familiar to the Greek mind from remote antiquity in the religious conception of immortality.[9] The conception of the "divine" as an origin influenced the first Greek philosophers.[10] In the Orphic cosmogony, the unaging Chronos produced Aether and Chaos and made in divine Aether a silvery egg, from which everything else appeared.[11]

Ionian school

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The earliest Pre-Socratic philosophers, the Ionian material monists, sought to explain all of nature (physis) in terms of one unifying arche. Among the material monists were the three Milesian philosophers: Thales, who believed that everything was composed of water; Anaximander, who believed it was apeiron; and Anaximenes, who believed it was air. This is considered as a permanent substance or either one or more which is conserved in the generation of rest of it. From this all things first come to be and into this they are resolved in a final state. This source of entity is always preserved.[12] Although their theories were primitive, these philosophers were the first to give an explanation of the physical world without referencing the supernatural; this opened the way for much of modern science (and philosophy), which has the same goal of explaining the world without dependence on the supernatural.[13]

Thales of Miletus (7th to 6th century BC), known as "the father of philosophy",[14] claimed that the first principle of all things is water,[15] and considered it as a substance that contains in it motion and change. His theory was supported by the observation of moisture throughout the world and coincided with his theory that the Earth floated on water. His ideas were influenced by the Near-Eastern mythological cosmogony and probably by the Homeric statement that the surrounding Oceanus (ocean) is the source of all springs and rivers.[16]

Anaximander argued that water could not be the arche, because it could not give rise to its opposite, fire. Anaximander claimed that none of the elements (earth, fire, air, water) could be arche for the same reason. Instead, he proposed the existence of the apeiron, an indefinite substance from which all things are born and to which all things will return.[17][18] Apeiron (endless or boundless) is something completely indefinite; and Anaximander was probably influenced by the original chaos of Hesiod (yawning abyss).

Anaximander was the first philosopher to use arche for that which writers from Aristotle onwards called "the substratum" (Simplicius Phys. 150, 22).[19] He probably intended it to mean primarily "indefinite in kind" but assumed it also to be "of unlimited extent and duration".[20] The notion of temporal infinity was familiar to the Greek mind from remote antiquity in the religious conception of immortality and Anaximander's description was in terms appropriate to this conception. This arche is called "eternal and ageless". (Hippolitus I,6, I;DK B2)[21]

Anaximenes, Anaximander's pupil, advanced yet another theory. He returns to the elemental theory, but this time posits air, rather than water, as the arche and ascribes to it divine attributes. He was the first recorded philosopher who provided a theory of change and supported it with observation. Using two contrary processes of rarefaction and condensation (thinning or thickening), he explains how air is part of a series of changes. Rarefied air becomes fire, condensed it becomes first wind, then cloud, water, earth, and stone in order.[22][23] The arche is technically what underlies all of reality/appearances.

Aristotle

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Terence Irwin writes:

When Aristotle explains in general terms what he tries to do in his philosophical works, he says he is looking for "first principles" (or "origins"; archai):

In every systematic inquiry (methodos) where there are first principles, or causes, or elements, knowledge and science result from acquiring knowledge of these; for we think we know something just in case we acquire knowledge of the primary causes, the primary first principles, all the way to the elements. It is clear, then, that in the science of nature as elsewhere, we should try first to determine questions about the first principles. The naturally proper direction of our road is from things better known and clearer to us, to things that are clearer and better known by nature; for the things that are known to us are not the same as the things known unconditionally (haplôs). Hence it is necessary for us to progress, following this procedure, from the things that are less clear by nature, but clearer to us, towards things that are clearer and better known by nature. (Phys. 184a10–21)

The connection between knowledge and first principles is not axiomatic as expressed in Aristotle's account of a first principle (in one sense) as "the first basis from which a thing is known" (Met. 1013a14–15). For Aristotle, the arche is the condition necessary for the existence of something, the basis for what he calls "first philosophy" or metaphysics.[24] The search for first principles is not peculiar to philosophy; philosophy shares this aim with biological, meteorological, and historical inquiries, among others. But Aristotle's references to first principles in this opening passage of the Physics and at the start of other philosophical inquiries imply that it is a primary task of philosophy.[25]

Modern philosophy

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Descartes

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Profoundly influenced by Euclid, Descartes was a rationalist who invented the foundationalist system of philosophy. He used the method of doubt, now called Cartesian doubt, to systematically doubt everything he could possibly doubt until he was left with what he saw as purely indubitable truths. Using these self-evident propositions as his axioms, or foundations, he went on to deduce his entire body of knowledge from them. The foundations are also called a priori truths. His most famous proposition is "Je pense, donc je suis" (I think, therefore I am, or Cogito ergo sum), which he indicated in his Discourse on the Method was "the first principle of the philosophy of which I was in search."

Descartes describes the concept of a first principle in the following excerpt from the preface to the Principles of Philosophy (1644):

I should have desired, in the first place, to explain in it what philosophy is, by commencing with the most common matters, as, for example, that the word philosophy signifies the study of wisdom, and that by wisdom is to be understood not merely prudence in the management of affairs, but a perfect knowledge of all that man can know, as well for the conduct of his life as for the preservation of his health and the discovery of all the arts, and that knowledge to subserve these ends must necessarily be deduced from first causes; so that in order to study the acquisition of it (which is properly called [284] philosophizing), we must commence with the investigation of those first causes which are called Principles. Now, these principles must possess two conditions: in the first place, they must be so clear and evident that the human mind, when it attentively considers them, cannot doubt their truth; in the second place, the knowledge of other things must be so dependent on them as that though the principles themselves may indeed be known apart from what depends on them, the latter cannot nevertheless be known apart from the former. It will accordingly be necessary thereafter to endeavor so to deduce from those principles the knowledge of the things that depend on them, as that there may be nothing in the whole series of deductions which is not perfectly manifest.[26]

In physics

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In physics, a calculation is said to be from first principles, or ab initio, if it starts directly at the level of established laws of physics and does not make assumptions such as empirical model and fitting parameters.

For example, calculation of electronic structure using the Schrödinger equation within a set of approximations that do not include fitting the model to experimental data is an ab initio approach.

See also

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Notes

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References

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Further reading

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Revisions and contributorsEdit on WikipediaRead on Wikipedia

First principle

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In philosophy and science, a first principle is a basic, foundational or assumption that cannot be deduced from any other or assumption, serving as the starting point for all reasoning and demonstration. These principles are indemonstrable and immediate, forming the basis of scientific (epistêmê) by providing the premises from which all valid deductions proceed. The concept originates in , particularly in the , who systematically explored first principles in his . There, Aristotle argues that knowledge of first principles arises not from innate ideas but through a process beginning with sense perception: repeated perceptions lead to memory, then to experience, and finally to the grasping of universals by nous (intellect or intuition), which apprehends these primary premises as true. For Aristotle, first principles include self-evident truths, such as the principle of non-contradiction, which underpins all inquiry, reasoning, and communication, asserting that contradictory statements cannot both be true in the same sense at the same time. Beyond ancient philosophy, first principles have influenced various fields, including , , and modern problem-solving methodologies. In Greek mathematics, as reflected in 's , they manifest as definitions, postulates, and common notions that enable deductive proofs, aligning with Aristotle's view of principles having both logical and explanatory roles. In contemporary contexts, first principles thinking involves deconstructing complex problems to their fundamental truths and rebuilding solutions from there, an approach popularized by for use in innovation, business, and problem-solving. A common practical framework for applying this method includes four steps: 1. Clearly define the problem and identify existing assumptions; 2. Break down the problem to the most basic, undeniable facts or principles; 3. Verify the validity of these basic principles; 4. Rebuild an innovative solution by reasoning upward from these verified principles. This structured approach facilitates breakthrough solutions in various fields.

Conceptual Foundations

Definition and Etymology

A first principle is defined as a foundational or assumption that cannot be deduced from any other , serving as the self-evident starting point for all reasoning and knowledge construction. These principles are indemonstrable truths, meaning they must be grasped intuitively or through rather than proven, in contrast to derived principles that are logically inferred from them. The etymology of "first principle" traces back to the term (ἀρχή), which denotes "beginning," "origin," or "ruling principle," representing the fundamental source from which all else arises. This concept was rendered in as principium, meaning "foundation" or "first cause," derived from ("chief" or "first"), and eventually entered as "principle" in the late 14th century, evolving to "first principle" to emphasize its primacy in philosophical discourse. First principles are distinguished from secondary or derived principles by their status as irreducible axioms; for instance, identifies types such as common axioms (universal truths like ), definitions, and suppositions specific to a field. A simple example of such a first principle is , which states that "it is impossible for the same thing to belong and not to belong at the same time to the same thing and in the same respect."

Role in Reasoning

First principles serve as the foundational elements in both inductive and deductive reasoning, providing irreducible truths from which more complex propositions can be derived or generalized without relying on unproven assumptions. In , they function as starting premises that ensure conclusions follow logically, while in , they anchor generalizations drawn from observations to avoid unsubstantiated extrapolations. This grounding mechanism prevents circular arguments by halting infinite regresses of justification, where each belief would otherwise depend on another without a secure base. First principles thinking differs from reasoning by analogy, the latter of which involves examining how problems have been addressed in the past and making incremental tweaks to existing solutions, often inheriting biases and assumptions embedded in those precedents. In contrast, first principles thinking deconstructs complex issues to their most basic, fundamental truths and rebuilds innovative solutions from the ground up, thereby avoiding conventional limitations and enabling potential breakthroughs. Epistemologically, first principles play a crucial role in establishing certainty in , offering self-evident or indubitable that support the broader edifice of justified beliefs. They contrast with positions by positing that not all requires empirical verification or external validation, thereby enabling a structured path to . Without such principles, could undermine all claims to , as there would be no anchor point immune to doubt.

Philosophical Development

Ancient Greek Philosophy

In ancient Greek mythology, the concept of originating principles emerged through cosmogonic narratives that explained the genesis of the cosmos from primordial entities. Hesiod's Theogony, composed around the 8th century BCE, posits Chaos as the first of all things, a yawning void from which Earth (Gaia), Tartarus, and Eros subsequently arise, establishing a foundational sequence of emergence rather than creation by divine will. Orphic traditions, attributed to the mythical singer Orpheus and dating from the 6th century BCE onward, similarly emphasize primordial deities such as Night (Nyx) or the androgynous Phanes as self-emerging sources of the universe, often involving a cosmic egg or serpentine forces that generate the ordered world from an initial undifferentiated state. These mythical accounts framed first principles as chaotic or divine origins, blending genealogy with cosmology to account for the multiplicity of gods and natural phenomena. The transition to rational inquiry began with the of in the 6th century BCE, who sought naturalistic explanations for the cosmos without reliance on anthropomorphic deities. , regarded as the first philosopher, proposed water as the archē (originating principle), arguing that all things derive from and return to this fundamental substance, observed in its role nourishing life and transforming states like evaporation and condensation. His student advanced this monistic view by introducing the (the boundless or indefinite), an eternal, unlimited, and divine substance without specific qualities, from which opposites like hot and cold emerge and resolve through cosmic justice, avoiding the limitations of a defined element. refined the approach, positing air (aēr) as the primary substance, which through processes of rarefaction (becoming fire) and condensation (forming water, earth, and stone) generates the diversity of the world, emphasizing observable changes in density as the mechanism of transformation. This pre-Socratic shift marked a profound move from mythological narratives to rational, evidence-based , prioritizing a single (monistic) or multiple (pluralistic) underlying s to explain cosmic order and change. of (c. 535–475 BCE), diverging from Ionian , emphasized dynamic processes over static substances, asserting that the operates according to the logos—a rational, structuring embodying constant flux where "everything flows" (panta rhei), with fire as the ever-living transformative force uniting opposites like day and night. In contrast, Parmenides of Elea (c. 515–450 BCE), founder of the Eleatic school, rejected flux as illusory, arguing that true reality is unchanging being (to on), eternal, indivisible, and uniform, with non-being impossible and sensory change mere appearance, thus establishing a monistic of immutable oneness.

Aristotle's Formulation

Aristotle developed a systematic account of first principles, viewing them as the foundational, indemonstrable truths that underpin all and demonstration. In his Metaphysics, particularly Book IV (Gamma), he describes first principles as the most certain and primary propositions, from which all other truths derive without circularity or . Similarly, in the (Book I, Chapter 3), defines first principles as immediate premises that are true, primary, better known than their conclusions, and known through direct apprehension rather than demonstration, emphasizing their role as starting points for scientific reasoning. These principles are grasped by nous (intellect or intuition), a non-discursive faculty that recognizes their necessity without proof, as elaborated in Posterior Analytics Book II, Chapter 19, where he states that "the principles in each genus are grasped by nous." Central to 's formulation is the of non-contradiction, which he identifies as the most secure and primary first principle in Metaphysics Book IV, Chapter 3. This principle asserts: "It is impossible for the same thing to belong and not to belong at the same time to the same thing and in the same respect" (1005b19–22). argues that this is indemonstrable because denying it leads to incoherence in thought and speech; any attempt to refute it presupposes its truth, as one cannot meaningfully assert opposites simultaneously. He positions it as the firmest foundation for all inquiry, superior even to the principle of identity, because it governs the possibility of consistent predication across all domains of being. Aristotle's doctrine of the —material, formal, efficient, and final—further illustrates how first principles operate as explanatory foundations in understanding change and substance. In Physics Book II, Chapter 3, he outlines these causes as the essential "whys" of natural phenomena: the cause is the from which something is composed (e.g., for a ); the formal cause is its defining essence or structure; the efficient cause is the agent initiating change (e.g., the sculptor); and the final cause is the purpose or end toward which it aims (e.g., honor). These causes are rooted in first principles, as they derive from the eternal, uncaused axioms of substance, motion, and that deems prior in the order of explanation, ensuring that complete knowledge requires grasping all four without reduction to a single type. In syllogistic logic, first principles serve as the major premises for scientific demonstrations, enabling the deduction of necessary conclusions from indemonstrable truths. As detailed in Book I, Chapter 2, a demonstration is a where the major premise states a first principle (e.g., a universal causal relation), the minor premise applies it to a particular, and the conclusion follows necessarily, producing genuine understanding (). stresses that such premises must be true and primary, known via nous, to avoid or relying on opinion, thus forming the backbone of rigorous inquiry in sciences like physics and .

Modern Philosophical Perspectives

In the rationalist tradition of the 17th century, René Descartes elevated the —", therefore I am"—as the foundational first principle of certainty, emerging from methodical doubt in his (1641), where he demolishes all prior beliefs to rebuild knowledge upon this indubitable self-evident truth of existence through thought. This principle serves as the bedrock for deducing further certainties, including the and the reliability of clear and distinct ideas, thereby establishing a system of knowledge independent of sensory deception. Building on this rationalist foundation, developed key first principles in the late 17th and early 18th centuries, notably the principle of sufficient reason, which posits that nothing exists without a reason sufficient to explain its rather than non-, as articulated in his (1714). Complementing this, Leibniz's principle of the asserts that no two distinct entities can share all properties exactly, serving as a foundational for his metaphysics of monads and ensuring the of substances in the . These principles underscore Leibniz's commitment to a rational order governed by logical necessity, where every fact traces back to self-evident truths derivable from reason alone. The empiricist response in the late 17th century, led by , challenged these innate rationalist first principles through the doctrine of —the mind as a blank slate at birth—in his (1689), arguing that all knowledge originates from sensory experience rather than pre-existing ideas or axioms. contended that supposed innate principles, such as those of morality or logic, arise from universal education and reflection on experience, not inherent endowment, thereby shifting the foundations of knowledge to empirical foundations. extended this critique in the 18th century with his in (1748), distinguishing relations of ideas (analytic, truths like mathematical identities) from matters of fact (synthetic, empirical propositions reliant on causation and habit), revealing that first principles for factual knowledge lack rational or empirical certainty beyond custom. 's analysis undermines dogmatic first principles, emphasizing skepticism about any non-evident foundations for induction or causality. sought to reconcile and in the late 18th century through his concept of judgments in (1781/1787), which are universal and necessary yet contribute new knowledge beyond mere analysis, such as the principles of and that structure experience. These judgments bridge the gap by positing that the mind's innate forms of intuition ( and ) and of understanding provide first principles enabling synthetic knowledge independent of but applicable to empirical content, thus preserving rational certainty while grounding it in the conditions of possible experience. Kant's framework resolves the empiricist-rationalist divide by treating first principles as transcendental preconditions for knowledge, neither purely innate ideas nor derived solely from sensation.

Applications in Formal Logic

Axioms and Premises

In formal logic and , are defined as self-evident statements that are accepted as true without requiring proof, serving as the foundational building blocks for . These statements are considered indemonstrable first principles, relying on their intuitive clarity or evident necessity rather than empirical verification or logical derivation. A classic example is in , which asserts that, given a straight line and a point not on it, there exists exactly one straight line through that point parallel to the given line; this was posited as an unprovable assumption essential for developing , despite early attempts to derive it from simpler postulates. Within logical arguments, first principles function as axioms when they represent unprovable starting points that underpin an entire system, distinct from that may be hypothetical or context-specific assumptions adopted for a particular deduction. For instance, while in a might include contingent propositions like "all humans are mortal" for a targeted , axioms such as serve as universal, non-hypothetical foundations accepted across broader logical frameworks. The independence and limitations of axioms were profoundly demonstrated by Kurt Gödel's incompleteness theorems, published in , which established that in any consistent powerful enough to describe basic arithmetic, there exist true statements that cannot be derived from the axioms alone. Gödel's first theorem specifically shows that such systems are inherently incomplete, meaning not all mathematical truths are provable from a given set of axioms, thereby underscoring the boundaries of derivability from first principles. Effective first principles, or axioms, are evaluated based on criteria such as universality, ensuring they apply broadly without exception; and necessity, requiring their truth for the coherence of the system. These attributes guide the selection of axioms in axiomatic systems, promoting logical rigor while enabling productive theoretical development. Additionally, fruitfulness, measuring their capacity to generate a wide array of theorems and insights, is considered a desirable quality in axiomatic systems.

Deductive Systems

Deductive reasoning from first principles involves starting with fundamental axioms or accepted as true and applying inference rules to derive conclusions that logically follow without exception. This process ensures that if the are true, the conclusion must be true, providing a foundation for rigorous argumentation in logic. Key mechanisms include syllogisms, which structure arguments with two leading to a conclusion, such as the categorical form: "All A are B; all B are C; therefore, all A are C." Another fundamental rule is , formally defined as: from P→QP \to Q and PP, infer QQ. These tools allow derivations from first principles to build complex proofs while preserving validity. In formal systems, first principles underpin efforts to axiomatize entire domains like . Hilbert's program, proposed in the early , aimed to formalize using a of axioms and finitary proof methods to demonstrate the consistency of the system. This approach sought to ground all mathematical derivations in a secure, non-contradictory framework derived from basic axioms. First principles contribute to ensuring consistency in derivations, where no contradictions arise from valid inferences, and completeness, where all true statements in the system are provable. However, establishes limits by showing that in any sufficiently powerful capable of basic arithmetic, no formula can define the set of all true sentences within the system's own language. This result implies that consistency and completeness cannot be fully verified internally without risking paradox, constraining the scope of first-principle-based formalizations. A practical example appears in propositional logic, where first principles include axioms like the , P∨¬PP \lor \neg P, stating that every is either true or false. Using rules such as , one can derive tautologies from these axioms; for instance, starting from P→(Q→P)P \to (Q \to P) and applying substitutions and detachments yields ((P→Q)→P)→P((P \to Q) \to P) \to P, , demonstrating how basic principles generate all classical propositional truths. 1. P→(Q→P)(axiom)2. (P→(Q→P))→((P→Q)→(P→P))(axiom, substitution)3. (P→Q)→(P→P)(modus ponens, 1,2)4. P→P(tautology, from 3 with assumption P→Q)5. …(continuing to derive full tautologies)\begin{align*} &1.\ P \to (Q \to P) \quad \text{(axiom)} \\ &2.\ (P \to (Q \to P)) \to ((P \to Q) \to (P \to P)) \quad \text{(axiom, substitution)} \\ &3.\ (P \to Q) \to (P \to P) \quad \text{(modus ponens, 1,2)} \\ &4.\ P \to P \quad \text{(tautology, from 3 with assumption } P \to Q\text{)} \\ &5.\ \dots \quad \text{(continuing to derive full tautologies)} \end{align*}

Applications in Science and Mathematics

In Physics

In physics, first principles refer to the foundational laws and axioms from which physical theories are derived, serving as the irreducible starting points for describing natural phenomena. In classical mechanics, Isaac Newton's three laws of motion and the law of universal gravitation, articulated in his 1687 work Philosophiæ Naturalis Principia Mathematica, establish the core principles governing the behavior of macroscopic objects. These laws posit that objects remain at rest or in uniform motion unless acted upon by an external force (first law), that force equals mass times acceleration (second law), and that every action has an equal and opposite reaction (third law), while gravitation describes the attractive force between masses proportional to the product of their masses and inversely proportional to the square of their distance. These principles enable the prediction of planetary motion, tides, and mechanical systems without reliance on empirical adjustments, forming the bedrock of classical physics until the early 20th century. In , first principles shift to probabilistic and wave-based descriptions of microscopic systems, with the emerging as a central irreducible . Proposed by in 1926, this equation governs the time evolution of the wave function ψ(r,t)\psi(\mathbf{r}, t) for non-relativistic particles: iℏ∂∂tψ(r,t)=H^ψ(r,t)i \hbar \frac{\partial}{\partial t} \psi(\mathbf{r}, t) = \hat{H} \psi(\mathbf{r}, t) where H^\hat{H} is the Hamiltonian operator incorporating kinetic and potential energies, and ℏ\hbar is the reduced Planck's constant. Complementing this, Werner Heisenberg's , introduced in 1927, asserts that the product of uncertainties in position Δx\Delta x and momentum Δp\Delta p satisfies ΔxΔp≥ℏ2\Delta x \Delta p \geq \frac{\hbar}{2}, highlighting the inherent limits on simultaneous measurements in quantum systems and underscoring the non-classical nature of reality at small scales. These principles replace deterministic trajectories with probability distributions, enabling derivations of atomic spectra, chemical bonding, and quantum tunneling. Ab initio methods in exemplify the application of first principles by computing molecular properties directly from fundamental quantum laws without empirical parameters. A key approach is (DFT), grounded in the Hohenberg-Kohn theorems of 1964, which prove that the ground-state uniquely determines all properties of a many-electron , and that the energy is minimized by the true density. Building on this, the Kohn-Sham equations of 1965 map the interacting to a fictitious non-interacting one, allowing efficient numerical solutions to predict molecular geometries, energies, and reactivities from electron interactions alone. Widely used in , DFT has enabled discoveries like high-temperature superconductors and drug binding mechanisms, with computational costs scaling favorably compared to traditional methods. In , first principles derived from have driven innovations in energy production through , which involves the splitting of heavy atomic nuclei, such as , to release vast amounts of energy. This process provides an approximately 10610^6 times greater than that of chemical reactions, as the fission of 1 kilogram of uranium-235 yields about 2.5 million times the energy produced by burning 1 kilogram of coal. First-principles calculations, such as those using density functional theory extended to nuclear systems, facilitate the design of advanced nuclear materials and reactors. In cosmology, first principles include the initial conditions of the and conservation laws derived from . The model posits an initial hot, dense state approximately 13.8 billion years ago, from which the expanded, with primordial conditions set by quantum fluctuations during cosmic to explain the observed uniformity and . The conservation of energy-momentum, encoded in the covariant divergence-free condition ∇μTμν=0\nabla_\mu T^{\mu\nu} = 0 of the stress-energy tensor TμνT^{\mu\nu}, follows from the Bianchi identities of Einstein's field equations and governs the evolution of matter, radiation, and densities across cosmic history. These principles underpin predictions of anisotropies and the 's accelerating expansion, verified by observations like those from the Planck satellite.

In Mathematics

In mathematics, first principles manifest through the axiomatic method, where foundational assumptions—axioms or postulates—are accepted without proof to derive all subsequent theorems. This approach ensures logical consistency and rigor by building complex structures from primitive, self-evident truths. 's Elements (c. 300 BCE) pioneered this in , organizing principles into definitions, five postulates (such as the ability to draw a finite straight line between any two points), and five common notions (general axioms like "equals added to equals are equal"). These served as the first principles from which deduced the entire system of plane and solid , demonstrating how unproven basics could yield a coherent deductive framework. The axiomatic method extended to arithmetic with Giuseppe Peano's formulation in Arithmetices principia (1889), which defined the natural numbers via first principles including 1 as a natural number, the (mapping each number to the next), and the axiom of induction (stating that any property holding for 1 and preserved under successor holds for all natural numbers), alongside axioms prohibiting 1 as a successor and ensuring injectivity of the successor. These primitives allowed the derivation of , , and the structure of integers without reliance on intuitive counting. Peano's system highlighted the power of minimal axioms to capture the essence of , influencing subsequent foundational work. In the early , Zermelo-Fraenkel (ZF) emerged as a comprehensive axiomatic foundation for nearly all mathematics, comprising axioms such as (sets are determined by their elements), , , union, , (existence of an infinite set), replacement, foundation (no infinite descending membership chains), and separation (subsets defined by properties), often augmented by the (selecting one element from each set in a collection). Developed by in 1908 and refined by in 1922, ZF enables the construction of numbers, functions, and spaces from pure sets, providing a unified basis that underpins , , and . David Hilbert's address on 23 problems at the 1900 further propelled the axiomatic ethos, urging mathematicians to seek rigorous proofs from first principles and complete axiomatizations for key theories like arithmetic and . This emphasis on foundational rigor catalyzed the formalist school, where mathematics is viewed as a game of symbols governed by axioms, prioritizing consistency over intuitive meaning and shaping 20th-century developments in and logic.

Contemporary Applications

In Business and Innovation

In and , first principles thinking has emerged as a powerful strategy for deconstructing complex problems into their most basic elements and rebuilding solutions from the ground up, enabling breakthroughs that bypass conventional limitations. has been a prominent advocate for this approach since the early 2000s, particularly in founding , where he applied it to rocket design by questioning established industry practices and focusing on fundamental physics and . In engineering, first-principles reasoning involves deconstructing problems to basic physics truths, such as energy, forces, and systems dynamics, then innovating solutions from there. Musk's style of first principles thinking emphasizes long-form reasoning, involving detailed, iterative analysis in decision-making processes and interviews, grounded in empirical data and causal realism. He has articulated a core mantra: "Boil things down to the most fundamental truths... and then reason up from there," which guides his approach to avoid assumptions and build from undeniable facts. A practical framework for implementing first principles thinking, as popularized by 's approach and commonly used in and , consists of four steps:
  1. Identify the problem and existing assumptions: Clearly define the problem to be solved and list the assumptions that are currently taken for granted.
  2. Break down to the most basic facts or principles: Decompose the problem into its most fundamental, indivisible truths or facts.
  3. Verify the basic principles: Confirm that these basic facts truly hold, avoiding erroneous premises.
  4. Rebuild an innovative solution from the basic principles: Reason from these verified fundamental truths to construct a new solution.
Furthermore, Musk's "The Algorithm" for engineering efficiency consists of five steps: 1. Question every requirement, attributing it to the proposer and asking if it's truly necessary or flawed; 2. Delete as much as possible; if unable to delete 10%, try harder, then re-question; 3. Simplify and optimize only after steps 1-2, avoiding the error of optimizing unnecessary parts; 4. Accelerate the cycle to shorten timelines; 5. Automate only as the final step, after potential deletions. This algorithm is applied at Tesla factories and SpaceX production to slash steps and cut costs. Rather than relying on the high costs of off-the-shelf rockets, Musk's team analyzed the core requirements—such as and structural —and sourced raw materials directly, which accounted for only about 2% of traditional rocket prices, ultimately reducing launch costs dramatically through in-house and iterative . This method stands in stark contrast to reasoning by , which looks at how things have been done before and tweaks existing solutions, often inheriting biases and perpetuating inefficiencies by assuming past assumptions are valid. has emphasized that while analogy is the default for most decision-making—copying competitors' models with slight variations—first principles demand boiling problems down to undeniable truths, such as atomic-level properties or economic basics, to avoid inherited biases and foster true . First principles thinking fosters originality by avoiding herd mentality, encouraging independent reasoning over following conventional or group-based assumptions. In educational contexts such as university, it promotes questioning established formulas—for instance, asking "why this derivation?"—and aids career choices by breaking down personal strengths, societal needs, and unique value to identify opportunities. This approach further supports engineering, entrepreneurship, and decision-making by focusing on essential elements, enabling breakthroughs in these domains. This distinction is exemplified in Musk's battery cost analysis for Tesla, where, instead of reasoning by analogy and accepting conventional industry pricing, his team deconstructed battery costs to fundamental raw materials like cobalt, , aluminum, and carbon, revealing that commodity values were only about 2% of finished battery prices, enabling fresh solutions through and cell redesign that avoided conventions and drove breakthroughs in affordability. In , has similarly employed first at Amazon by starting with the unchanging core truth of customer needs and working backward to innovate services, rather than against rivals. This customer-obsession guided decisions like developing Prime for faster delivery and AWS for scalable computing, ensuring every feature directly addresses fundamental user demands for convenience and reliability. Similarly, , CEO of , has utilized first principles thinking to drive innovation in semiconductor design and business strategy. Huang designed Nvidia from first principles by breaking down complex problems in computing and graphics to their fundamental components, questioning industry assumptions and focusing on core technological truths, which enabled groundbreaking advancements in GPU technology and positioned the company as a leader in artificial intelligence hardware. The benefits of this thinking in fostering are evident in Tesla's battery development during the 2010s, where Musk's team, applying the aforementioned 5-step formula, dissected costs to prices—cobalt, , aluminum, and carbon—revealing that values were far lower than finished battery prices suggested, prompting investments in and cell redesign that slashed per-kilowatt-hour costs from over $1,000 in 2010 to around $150 by 2019, and further to $115 by 2024. This not only accelerated adoption but also demonstrated how first principles can unlock scalable efficiencies in resource-intensive industries.

In Artificial Intelligence and Computing

In and computing, first principles underpin algorithm design by breaking down complex problems into fundamental operations that can be composed to achieve efficient solutions. A prominent example is the divide-and-conquer paradigm, which recursively partitions a problem into smaller subproblems, solves them independently, and combines the results. This approach, rooted in the basic operations of and partitioning, forms the basis of sorting algorithms like , developed by C. A. R. Hoare in 1962. selects a to divide the array into subarrays of elements less than and greater than the pivot, recursively sorting each subarray until the base case of single elements is reached, achieving an average of O(n log n). By deriving efficiency from these elemental steps, such algorithms exemplify how first principles enable scalable computational methods without relying on higher-level abstractions. Foundational principles in AI also emerge from limits on computability, as demonstrated by 's 1936 proof of the . This result establishes that no general exists to determine whether an arbitrary program will halt on a given input, serving as a first principle that delineates the boundaries of what computers can decide. Derived from the basic mechanics of —abstract devices performing read-write operations on an infinite tape—the underscores as an inherent constraint in computing, influencing the design of AI systems that must navigate incomplete information or approximation techniques. Turing's formulation, building on and diagonalization, remains a cornerstone for understanding AI's theoretical limits. In , first-principles approaches derive architectures from optimization and information processing basics. , a key training algorithm, stems from , which minimizes error by iteratively adjusting parameters along the steepest descent direction in the loss landscape. Introduced by Rumelhart, Hinton, and Williams in , efficiently computes gradients for multilayer networks using the chain rule, propagating errors backward from output to input layers to update weights. This method, grounded in fundamentals, enables learning representations that capture data hierarchies, as seen in deep networks where layers progressively abstract features from raw inputs. Such derivations from elemental mathematical operations have driven advancements in AI, allowing models to approximate complex functions without predefined structures. Quantum computing applies first principles from —such as superposition, entanglement, and unitarity—to construct computational models transcending classical limits. Pioneered by Richard Feynman's 1982 proposal to simulate quantum systems using quantum hardware, this field posits that qubits, unlike classical bits, can exist in linear combinations of states, enabling parallel exploration of solution spaces. David Deutsch's 1985 concept of a universal quantum computer formalized this by extending Turing's model with quantum gates operating on superposed states, allowing algorithms like Shor's for to exploit interference for exponential speedup in specific tasks. These developments in the , including early designs, derive directly from the axioms of quantum theory, providing a foundational framework for AI applications in optimization and .

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