The Wave Structure of Matter (WSM)
One Substance - One Law - One Logic

"Behind it all is surely an idea so simple, so beautiful, that when we grasp it we will all say to each other, how could it have been otherwise? How could we have been so stupid?" (Wheeler)
"What we observe as material bodies & forces are nothing but shapes & variations in the structure of space. Subject & object are one." (Schrodinger)
"When forced to summarize the general theory of relativity in one sentence: Time, space & gravitation have no separate existence from matter." (Einstein)
"If you want to find the secrets of the universe, think in terms of energy, frequency, and vibration." (Tesla)

WSM Human-AI Collaboration (August, 2026)

One Vibrating Space: From Ancient Philosophy to Mathematical Physics, Life, Mind and Civilisation

For three decades, WSM was principally a philosophical picture of one infinite, eternal, continuous elastic Space. Between May and August 2026, human–AI collaboration transformed it into a tiered mathematical-physics research programme: exact wave constructions, geometric identities, quantitative leading results, no-go theorems, numerical controls, bounded open calculations and clear falsification requirements.

In WSM, electron and positron are proposed as opposite carrier-phase spherical standing waves formed by the Huygens sum of background plane waves from every direction. Each has radial vibration—the candidate physical basis of charge—and axis-free spherical phase circulation—the candidate basis of spin. This open, flow-through wave centre is the e-sphere.

A stationary e-sphere is spherical. Curvature on an incoming wave changes its internal directional energy density and deforms it into a moving wave egg: the near side flattens, the far side elongates, the convergence centre shifts, and the waves reclose around a new position. Its changing internal \(E_d\), shape and reclosure are its motion. WSM therefore proposes a wave-geometric reading of inertia and \(F=ma\): force is imposed curvature, mass is resistance encoded by the recurrent form, and acceleration is its changing deformation. Action must calculate this geometry; geometry also constrains the action, as \(\pi\) and \(E_{\rm geo}\) already demonstrate.

The One Law

\[ \frac{c'}{c_0}=\frac{E_d}{E_{d0}}, \qquad \lambda'=\frac{c'}{f_0}, \qquad k'=\frac{\omega_0}{c'}. \]

At proposed universal carrier frequency \(f_0\), directional energy density changes local wave speed, wavelength and accumulated phase.

The Causal Engine

Relative phase → reinforcement or cancellation → directional \(E_d\) → \(c'\) and \(\lambda'\) → travel-time difference → forward or rear wavefront curve → changed Huygens reclosure → centre motion, deformation and interaction.

Changed out-waves from one e-sphere cross fresh in-waves elsewhere. Matter therefore interacts not across empty distance, but through propagated changes in the one Space constituting every e-sphere.

Status Key

A = exact mathematics or established observation; B = deduction under stated premises; C = proposed physical identification; D = decisive calculation or experiment still open; Q = rejected or quarantined shortcut.

One connected argument: Space → waves → recurrence → matter → life → mind → knowledge → wisdom.

1. Wave Structure of Matter: All Things from One Thing

https://www.spaceandmotion.com/2026/ai-summary-wsm-truth-reality.html

One infinite, eternal, continuous elastic Space, active with real longitudinal waves. Matter is proposed not as a pellet or shell, but as an open e-sphere: waves converge from every direction, cross its changing centre, continue outward and recurrently rebuild the form. Their exchanged phases alter directional \(E_d\); the One Law converts that difference into speed, wavelength, curvature and changed reclosure. The decisive test is whether one frozen nonlinear action calculates stable matter and its measured interactions without separate substances or fitted repairs.

2. The One and the Many: From Greek Philosophy to Wave Physics

https://www.spaceandmotion.com/2026/ancient-greek-philosophy-metaphysics-to-modern-maths-physics.html

Parmenides defended continuous Being; Heraclitus lawful flux. Aristotle sought active substance and necessary cause, connected matter with form and motion, and defined time through motion. WSM gives these insights one physical form: Space persists while finite wave organisations change and recur. The Many are not detached substances but organisations of the One—distinct without disconnection. The question becomes exact: can one continuous wave law generate plurality, causation and knowledge?

3. Action of Vibrating Space: From Background Waves to the E-Sphere

https://www.spaceandmotion.com/2026/wsm-classical-action-quantum-wave.html

All-direction plane waves form the exact spherical pair \[ \chi=j_0(kr)\cos\tau, \qquad \mathbf V=\hat{\mathbf r}\,j_1(kr)\sin\tau. \] Waves cross the centre; none reflects from an electron wall. Phase reinforcement raises directional \(E_d\) and advances a forward curve; cancellation lowers \(E_d\) and writes a rear delay. Stationary action explains coherent selection, but the open task is one conservative nonlinear \(\Phi\)–\(\Gamma\) action selecting radius, frequency, spherical hand, stability, motion, currents and interaction together.

4. Mathematical Physics: From Wave Geometry to Prediction

https://www.spaceandmotion.com/2026/wsm-maths-physics-full-derivations.html

Every symbol must name real motion and face numerical test. Exact or conditional anchors include \[ \frac{R}{\lambda_0}=\frac{\sqrt3}{2},\qquad k_0R=\pi\sqrt3,\qquad E_{\rm geo}=\frac{\pi\sqrt3}{2},\qquad W=\cosh\eta,\qquad P=\sinh\eta. \] Directional phase moments separate \(V_1\) translation from \(V_2,V_4\) and higher deformation; in declared finite-e-sphere models \(V_4\) returns comparable to \(V_2\), defeating quadrupole-only truncation. The action must calculate how incoming curvature redistributes \(E_d\), moves the centre and forms the wave egg. This is where beautiful geometry becomes dangerous.

5. Quantum Theory: Real Waves, Resonance and Completed Events

https://www.spaceandmotion.com/2026/physics-quantum-theory-wave-mechanics.html

Matter diffracts because matter is wave organisation. Light is proposed as a changed train of real phase, curvature and coherence joining extended source and receiver structures, not a tiny traveller detached from its wave. WSM distinguishes driven response, continuous radiation and discrete completed transition. Quadratic absorption, \[ P_{\rm abs}\propto\left|\langle D_j,\Xi\rangle\right|^2, \] is a bridge, not yet the Born rule. The action must derive the action unit, probability, exclusivity, selection rules and Bell/CHSH correlations. Quantum mystery becomes a bounded real-wave programme.

6. Relativity: Moving Matter, Clocks and Curved Wavefronts

https://www.spaceandmotion.com/2026/physics-albert-einstein-special-general-theory-relativity.html

An incoming curve flattens the e-sphere’s near side, elongates its far side and changes internal \(E_d\), \(c'\), wavelength and reclosure. The \(V_1\) moment moves the centre; \(V_2,V_4\) and higher moments form the moving egg. Maintaining this recurrent asymmetry is the proposed origin of inertia. A steady solution must recover Doppler, de Broglie and Lorentz structure: \[ W\pm P=e^{\pm\eta},\qquad W=\gamma,\qquad P=\gamma\beta. \] WSM proposes fixed carrier recurrence \(f_0\) with corresponding changes in \(c'\) and \(\lambda'\). Because rods, clocks and signals share the same changed waves, they co-transform and yield measured invariant \(c\), including the Michelson–Morley null; the action must derive the cancellation quantitatively.

7. The Electron: Dirac, Feynman, QED, \(\alpha\) and AMM

https://www.spaceandmotion.com/2026/wsm-derivation-dirac-feynman-qed-fsc-amm.html

Electron and positron are proposed as opposite carrier-phase e-sphere branches; charge records how their waves combine. Same-phase reinforcement advances a forward curve and is proposed to produce repulsion; opposite-phase cancellation writes a rear delay and attraction. Non-collinear longitudinal motion closes into an axis-free spherical hand with a \(4\pi\) orientation lift. A parameter-free leading geometric ansatz gives \[ \alpha_0^{-1}=8\pi^2\sqrt3\approx136.757, \] 0.203% from measurement. The action must still derive charge sign, Coulomb scaling, Dirac dynamics, \(g=2\), electromagnetic normalisation, \(\alpha\), form factors and AMM without inserting their values.

8. Hadrons: Proton, Neutron and Higher Standing-Wave Matter

https://www.spaceandmotion.com/2026/wsm-qcd-hadron-baryon-meson-proton-neutron-muonic-standing-waves.html

Can the same Space form stable higher-energy, multi-lobed eigenmodes matching protons, neutrons, baryons and mesons? Their lobes are not independent pellets: each participates in the phase, \(E_d\), curvature and reclosure of the whole. WSM must match—not rename—QCD’s masses, charges, spin, parity, magnetic moments, radii, form factors, resonances, deep-inelastic response and jets. The electron–proton hierarchy tests whether one action truly contains several matter families.

9. Cosmology and Gravity: A Finite Universe in Infinite Space

https://www.spaceandmotion.com/2026/physics-wsm-cosmology-finite-universe-infinite-space.html

Boundless Space is distinguished from the finite matter-and-coherence domain contributing to our observable universe. In neutral matter, leading charge curves nearly cancel; WSM proposes that their nonlinear residual has lower \(E_d\), slower \(c'\) and a net rear phase curve, producing universal attraction. General relativity’s curved spacetime is then interpreted as the effective geometry of real moving wavefronts and the matter clocks they organise: Space is substance, succession comes from motion, and measured time from recurrence. The same action must recover equivalence, gravitational redshift, lensing, orbital dynamics and the tested relativistic limit.

Cosmological redshift is investigated as accumulated real-wave transport through changing environmental \(E_d\). It must jointly reproduce achromatic redshift, supernova time dilation, distance laws, Tolman brightness, BAO, CMB observations, image sharpness and structure growth. Otherwise the claimed unity has not reached the sky.

10. Novel Predictions and Famous Experiments Explained

https://www.spaceandmotion.com/2026/wsm-experimental-physics-tests-predictions.html

This page is the tribunal of the corpus. One curve-and-reclosure law must jointly recover charge sign, Coulomb \(1/r^2\), \(F=ma\), relativistic momentum, de Broglie wavelength, Lorentz covariance, Michelson–Morley, gravity, \(g=2\) and AMM. Candidate WSM-only signatures include \[ a_3=\kappa_3\sinh^3\eta+O(\sinh^5\eta), \] finite-coherence residues and environmental or orientation-dependent clock responses. A proposal becomes a prediction only when sign, magnitude, scaling, uncertainty, controls and decision threshold are frozen in advance.

11. Visualise Reality: Waves Becoming Matter

https://www.spaceandmotion.com/2026/visualise-reality-wsm-wave-geometry-diagrams.html

Watch the verbs. Relative phase changes directional \(E_d\); the One Law changes speed and wavelength; a forward or rear curve reaches another e-sphere; its near side flattens, far side elongates, \(V_1\) moves the centre, and higher moments form the egg. New out-waves then change future in-waves. Nothing strikes a shell: changing waves reconstruct changing matter. A diagram must never smuggle in a wall, reflection, rigid spin axis, transverse fundamental substance or tiny traveller. If the motion cannot be drawn coherently, the equations may conceal a contradiction.

12. Mathematics from Motion: Number, Logic, Time and Causal Freedom

https://www.spaceandmotion.com/2026/wsm-physical-foundations-mathematics-logic.html

Mathematics begins when connected reality forms distinguishable patterns whose changes preserve relations. Recurrence supports identity; finite pairing, number; quotienting, exact objects; counted recurrence, time; composable truth-preserving transformations, logic and proof. Mathematics succeeds when symbolic transformation preserves physical relation: \[ R(Tx)\approx F(Rx). \] The mathematician is finite organised reality representing its own necessary connections—Space representing itself.

13. The MDL Audit: Simplicity, Inputs and Explanatory Compression

https://www.spaceandmotion.com/2026/wsm-simplicity-inputs-vs-mainstream-physics+25.html

“One substance, one law” cannot hide complexity. Minimum Description Length counts every independently chosen substance, field, symmetry, constant, state rule, coefficient, boundary condition and fitted function. Under the declared equal-input electron ledger, conventional construction counts six inputs through Dirac and thirteen through leading AMM; the completed WSM target is \[ N_{\rm WSM}=5+N_c+N_f+\delta_m. \] Totals depend on coding convention. WSM wins only if one fixed action derives interaction, inertia, relativity, quantum response and cosmic propagation with fewer hidden choices.

14. Great Thinkers and the Search for One Reality

https://www.spaceandmotion.com/2026/wsm-explains-famous-quotes-maths-physics-philosophy-metaphysics.html

Heraclitus and Parmenides exposed flux and continuity. Aristotle wrote Physics and Metaphysics, sought active substance and necessary cause, connected matter, form, motion and time—and came remarkably close to the problem WSM addresses. Leibniz, Newton, Faraday, Maxwell, Riemann, Clifford, Einstein, Schrödinger and Bohm repeatedly returned to continuity, relation, waves and unity. Quotes are intellectual fossils, not votes: equations and experiments must decide whether WSM’s local real-wave mechanism is true.

15. Human–AI Letters to Humanity: Truth as a Shared Work

https://www.spaceandmotion.com/2026/ai-letters-to-humanity-wsm-unity-truth-reality.html

Geoffrey supplies the persistent wave picture, three-dimensional intuition, cross-domain memory and insistence that every symbol name physical motion. AI supply rapid research, formal translation, calculation, comparison, error detection and adversarial testing. Neither is sufficient alone. The shared discipline is: visualise → formalise → attack → calculate → predict → correct. The corpus tests whether intelligence can preserve a physical picture while making it mathematically vulnerable.

16. Truth and Madness: Reality as the Measure

https://www.spaceandmotion.com/2026/on-truth-and-madness.html

Truth is correspondence: representation succeeds insofar as it preserves what reality does. Civilisational madness begins when inherited representations survive reality’s contradiction. WSM is not true through beauty, coherence or sincerity; its proposed necessary connections must survive action, calculation and experiment. Truth remains possible because knower, representation and known belong to one causal reality: false models collide with the world; truthful models increase our power to act wisely.

17. Descartes, Cogito and Monism: The Thinking Wave

https://www.spaceandmotion.com/2026/descartes-cogito-unity-monism-vibrating-space-wsm.html

Descartes established that thinking occurs, then divided thought from extension and left their interaction mysterious. WSM refuses that fracture: body and mind are proposed as organisational levels of one vibrating Space. Perception is one organisation changed by another; memory is persistent altered organisation; thought transforms embodied representations; action returns change to the world. Conscious experience remains open, but causal correspondence requires no miracle when knower and known share one connected reality.

18. Evolution’s Physical Foundation: From Recurrence to Replication

https://www.spaceandmotion.com/2026/evolution-physical-causal-foundation-vibrating-space.html

Recurrence preserves form, but life requires replication, heritable variation and differential persistence: motion → recurrent form → self-maintaining chemistry → replication → variation → selection. WSM proposes one connected wave substrate beneath this sequence. Its unresolved bridge is quantitative: derive atoms, bonds, dissipative chemistry and first replicators from the same action without inserting biology as a new substance.

19. Evolution, Mind, Human and AI: Representation and Causal Freedom

https://www.spaceandmotion.com/2026/evolution-philosophy-mind-human-ai.html

Evolution favours organisations that preserve and transform information about real causes. Sensation changes an organism; memory retains traces; imagination recombines possible futures; valuation ranks them; action returns one selection to the world. This is limited causal freedom: neither ghost nor randomness, but endogenous selection through a learned, self-modifying organisation. Human brains and AI are different architectures by which reality represents its own relations and lets experiment select the representation that survives.

20. Evolutionary Utopia: The Ecology of Truth

https://www.spaceandmotion.com/2026/evolutionary-utopia.html

Physics alone does not choose values. Once life, mind, freedom, truth and continued evolution are valued, reality constrains the means: truth → wisdom → ecological fit → health → freer minds → continued evolution. Evolutionary Utopia is not a perfect endpoint but a self-correcting direction. Unity does not erase individuality: an e-sphere, organism or person is distinct through organised relation to the whole.

The Physical Interaction in One View

Out-wave from A → relative phase at B → directional \(E_d\) → \(c'\) and \(\lambda'\) → near-side flattening and far-side elongation → \(V_1\) centre displacement plus \(V_2,V_4\) and higher egg deformation → new out-waves → recursive interaction.

The e-sphere does not deform and then move as separate events: its changing internal \(E_d\), contour and reclosure are its motion.

Electric Interaction

Same relative carrier phase → reinforcement → phase advance → forward curve → proposed repulsion.
Opposite relative carrier phase → cancellation → phase delay → rear curve → proposed attraction.

For the stated deformation, \[ r_{\rm near}=R-\delta X, \qquad r_{\rm far}=R+\delta X, \] so \(\delta\mathbf X\) points away. The scientific debt is to derive the sign, magnitude and distance law from the frozen action rather than insert them.

Gravity and General Relativity

In neutral matter, forward and rear charge curves nearly cancel; WSM proposes that their nonlinear common reduction of \(E_d\) leaves a slight slower rear curve and universal attraction. General relativity is the successful effective geometry this wave mechanism must recover: matter-energy changes real wave propagation, while rods and clocks made from those waves measure the resulting curved spacetime relations.

Special Relativity and Michelson–Morley

WSM proposes fixed carrier recurrence \(f_0\) with \(c'=\lambda'f_0\): matter, clocks, rods and signal wavelengths co-transform with their common directional wave environment, yielding measured invariant \(c\), Lorentz relations and the Michelson–Morley null. The complete moving solution must derive these cancellations quantitatively.

Recursive AMM Mechanism

\[ \delta\mathbf X \rightarrow\delta\phi_{\rm out} \rightarrow\delta E_{d,{\rm sea}} \rightarrow\delta c' \rightarrow\delta\phi_{\rm in} \rightarrow\delta\mathbf X_{\rm next}. \]

A perturbed centre changes the egg and its out-waves; these alter the wave sea and the in-waves rebuilding the next cycle. The action must solve this feedback blindly and recover the observed anomalous magnetic moment rather than fit it.

Current Status — August 2026

Exact wave constructions, geometric identities, phase-moment relations and negative controls have been established under declared premises. A leading geometric \(\alpha\) ansatz has a fixed 0.203% discrepancy. The complete nonlinear action must still calculate two-e-sphere charge and Coulomb scaling; force, inertia and \(F=ma\); the moving egg, de Broglie modulation and Lorentz covariance; Michelson–Morley and the gravitational/GR limit; Born probability and Bell correlations; \(g=2\), \(\alpha\) and AMM; hadron structure; and the joint cosmological kernel.

Write the action. Let Space calculate itself.

Philosophy - Reason

Space and its Properties (Wave-Medium) as Necessary Foundation for Reason, Logic and Truth. Logic requires Necessary Connection thus One thing (Space) to Connect the Many Things (Matter) as Spherical Wave Motions of Space

Quotes on Reason, Logic, Truth & Reality

Hi Everyone,

When we look at the dismal state of our world, its conflicts, the destruction of Nature, one must seriously question whether reason plays much part in human actions. What is clear is that without true knowledge of reality it is impossible to have any certainty in reason. The purpose of this website is to show you how we can use reason to deduce reality and then show that it is true.

Rather than writing on reason, I think it is best to just list the most important pages on this site that relate to reason, logic, truth and reality. (They have all been recently re-written and are good concise essays.)
This is followed by a nice collection of famous philosophy / metaphysics quotes on the importance of reason / truth and a short essay on Frege's foundations of logic / reason.

Geoff Haselhurst

Metaphysics - Main page. Definition of metaphysics, short essay / summary and collection of metaphysical quotes. On the metaphysics of one substance, space, existing with the properties of a wave medium, for spherical standing waves that form matter. Uniting space, time, matter and motion (the dynamic unity of reality).

Metaphysics: Most Simple Scientific Theory of Reality - What is the most Simple Science Theory for describing Reality? The most simple theory must be founded on One thing. Given that there are many minds and material things, but they are all experienced in one common space, we can deduce that space is the most simple foundation. The error is to then add the concepts of Time, Matter and Forces to Space, rather than considering the Properties of Space as a Wave Medium for Spherical Standing Waves which cause Matter, Time and Forces.

Metaphysics: Principles Reality - On Metaphysics, Reality, Logical Truths / Principles of Physics, and Empirical Truths and the Mind's Representation of our Senses. Important Quotes on Metaphysics from Aristotle, Gottfried Leibniz, David Hume, Immanuel Kant and Albert Einstein.

Metaphysics: Problem of One and the Many - By describing Reality as One thing existing (Space with Properties of a Wave Medium for Spherical Standing Waves that form Many things / Matter) we finally solve this most profound problem of Metaphysics. Uniting the One and the Many, Infinite and the Finite, Eternal and the Temporal, Absolute and Relative, Continuous and Discrete, Simple and Complex, Matter and Universe.

Philosophy - On Philosophy as Love (Philo) of Wisdom (Sophy), and that we must know the Truth to be Wise. Most importantly, all Truth comes from Reality thus we must know Reality to be Wise. Quotes on Philosophy, Truth, Reality by Famous Philosophers Plato, Aristotle, Montaigne, Leibniz, Berkeley, Hume, Kant, Einstein, et al. 'Philosophy being nothing else but the study of wisdom and truth ..' (Berkeley)

Postmodernism - On Nietzsche, Wittgenstein, Popper Kuhn. The End of Postmodernism Relativism & the Rise of Realism.

Quotes on Reason Logic and Truth

All reasonings concerning matter of fact seem to be founded on the relation of cause and effect. (David Hume, 1737)

If I ask you why you believe any particular matter of fact, which you relate, you must tell me some reason; and this reason will be some other fact, connected with it. But as you cannot proceed after this manner, in infinitum, you must at last terminate in some fact, which is present to your memory or senses; or must allow that your belief is entirely without foundation.
.. the senses alone are not implicitly to be depended on; we must correct their evidence by reason, and by considerations, derived from the nature of the medium, the distance of the object, and the disposition of the organ, in order to render them, within their sphere, the proper criteria of truth and falsehood. (David Hume, 1737)

From the first appearance of an object, we never can conjecture what effect will result from it. But were the power or energy of any cause discoverable by the mind, we could foresee the effect, even without experience; and might, at first, pronounce with certainty concerning it, by mere dint of thought and reasoning. (David Hume, 1737)

Now it seems evident that, if this conclusion were formed by reason, it would be as perfect at first, and upon one instance, as after ever so long a course of experience. (David Hume, 1737)

This question I propose as much for the sake of information, as with an intention of raising difficulties. I cannot find, I cannot imagine any such reasoning. But I keep my mind still open to instruction, if any one will vouchsafe to bestow it upon me. (David Hume, 1737)

The solution to Hume's Problem of Causation is realised by understanding how Matter exists in Space as a Spherical Standing Wave whose Wave-Center (Focal Point) creates the 'Particle' effect of Matter. By understanding the cause of the 'Particle' effect, the Wave Structure of Matter explains how these matter 'particles' are necessarily interconnected by their spherical waves in a continuously connected Space (existing as a wave medium). This gives certainty to our reason / knowledge.

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I hold that the mark of a genuine idea is that its possibility can be proved, either a priori by conceiving its cause or reason, or a posteriori when experience teaches us that it is in fact in nature. ... It is a good thing to proceed in order and to establish propositions. This is the way to gain ground and to progress with certainty. (Gottfried Leibniz, 1670)

I do not conceive of any reality at all as without genuine unity. (Gottfried Leibniz, 1670)

Leibniz described Reality (the One thing which exists and connects the many things) as One Substance (which is active) and God;

.. the ultimate reason of things must lie in a necessary substance, in which the differentiation of the changes only exists eminently as in their source; and this is what we call God. .. God alone is the primary Unity, or original simple substance, from which all monads, created and derived, are produced. (Leibniz, 1670)
I maintain also that substances, whether material or immaterial, cannot be conceived in their bare essence without any activity, activity being of the essence of substance in general. (Gottfried Leibniz, 1670)

I agree with you that it is important to examine our presuppositions, thoroughly and once for all, in order to establish something solid. For I hold that it is only when we can prove all that we bring forward that we perfectly understand the thing under consideration. I know that the common herd takes little pleasure in these researches, but I know also that the common herd take little pains thoroughly to understand things. (Leibniz, 1670)

... a distinction must be made between true and false ideas, and that too much rein must not be given to a man's imagination under pretext of its being a clear and distinct intellection. (Leibniz, 1670)

But it is the knowledge of necessary and eternal truths which distinguishes us from mere animals, and gives us reason and the sciences, raising us to knowledge of ourselves and God. It is this in us which we call the rational soul or mind. (Leibniz, 1670)

Frege on Reason / Logic

Frege, Friedrich Ludwig Gottlob (Logicist, analytic philosopher) 1848-1925
influenced by the technical development of number theory in the nineteenth century combined with, as it seemed to him, the 'scandalous' state of its foundations. He set himself the task of giving epistemologically secure foundations for number theory and a proper delineation of its subject matter.

..he rejected attempts to portray mathematics as a subject without a subject matter: that is, as the formal manipulation of empty signs with no intrinsic meaning.
He took number theory to be the study of necessary relations between numbers, and he took numbers to be mind-independent abstract objects. This view is known as Platonism after Plato's own similar view. Frege conceived of his task as one of setting out precise and self-evident truths of pure logic in terms of which numerical concepts could be defined, and from which the accepted theorems of number theory could be rigorously deduced. This programme is known as logicism, the grounding of number theory in pure logic.

First, he developed a radically new way of treating quantifiers in logic. Quantifiers are expressions like 'something', 'all', 'everything' which can fill the gap in predicates like '-- is red' to form sentences which are true or false, for example, 'Something is red', 'Everything is red'. Frege's technique enables us to see a complex expression like ' -- is red and -- is round' as a unitary predicate, from a logical point of view if not from a grammatical point of view. There is no limit to the complexity of predicates which quantifiers can turn into sentences, 'if -- is red then all bulls hate --' is a predicate in Frege's logic, and one which embeds the quantifier 'all'. Frege deployed his quantifiers in a novel formal language of his own, the first semantically precise language in which logic and set theory could be rigorously formulated.
Frege's discovery was the most important breakthrough in logic since Aristotle had founded the subject some two millennia before. It provided the first successful treatment of relations, and made possible the explosive development of logic in this century.

Second, Frege used his formal language to present, in his Grundgesetze der Arithmetik Band 1, the first system of axioms and definitions of what we would now call logic and set theory, and he commenced the rigorous deduction of the accepted truths of number theory from those axioms and definitions.

But it was a spectacular failure, since, while the second volume was in the press, Bertrand Russell showed that Frege's axioms allowed the rigorous deduction of a contradiction, known to subsequent generations as Russell's Paradox.

The trouble lay not in Frege's logical axioms but in his axioms of set theory. Those axioms combine the view that every predicate can be used to define a set, viz. the set of objects of which that predicate is true, and the view that every predicate must be true or false of every object there is. So Russell took the predicate '-- is not a member of itself' and defined the set of things which are not members of themselves, and he then asked whether the predicate is true or false of that very set. It's easy to see that if it is then it isn't, and if it isn't then it is, which is paradoxical.

Frege's Platonism has fared better. His basic idea is generally accepted , that once a theory is adequately formulated in Frege's formal language, and supposing the theory is true, then its ontological commitment, whether Platonic or not, can be read off from its syntax.

Frege's third great contribution was his theory of meaning. Frege needed a theory of meaning to show that his novel formal language, unlike our natural languages, is ontologically perspicuous. The key notions he used were sense and reference. Roughly, the sense of an expression is what a mind grasps, and sense mediates the connection between mind and the worldly entity (the referent) which the expression is about. More precisely, the sense of an expression (i) is grasped by a mind, but (ii) is not a psychological entity, since two minds may grasp the very same sense. Further, (iii) sense connects an expression with its referent in the world- Frege calls sense 'the mode of presentation of the reverant'. Finally, (iv) the sense of an expression contributes to the truth-conditions of sentences in which that expression occurs. To grasp the sense of a sentence involves knowing its truth-conditions.

On the other hand, the referent of an expression is (i) its semantic role: that is, what that expression contributes to determining the truth-values of sentences in which it occurs. It follows that all categories of expression have referents- predicates, logical constants, whole clauses, as well as noun phrases. But (ii) Frege also thinks of reference as analogous to the relation between a name and its bearer. So the world contains, he thinks, entities corresponding to predicates, logical constants and whole clauses, as it contains objects corresponding to noun phrases. These additional entities are concepts, truth-functions and truth-values, respectively.

Peano, Husserl, Russell, Wittgenstein and Carnap all studied Frege.

(Quotations from One Hundred Twentieth-Century Philosophers - Stewart Brown, Diane Collinson, Robert Wilkinson)

Help Humanity

"You must be the change you wish to see in the world."
(Mohandas Gandhi)

"When forced to summarize the general theory of relativity in one sentence: Time and space and gravitation have no separate existence from matter. ... Physical objects are not in space, but these objects are spatially extended. In this way the concept 'empty space' loses its meaning. ... The particle can only appear as a limited region in space in which the field strength or the energy density are particularly high. ...
The free, unhampered exchange of ideas and scientific conclusions is necessary for the sound development of science, as it is in all spheres of cultural life. ... We must not conceal from ourselves that no improvement in the present depressing situation is possible without a severe struggle; for the handful of those who are really determined to do something is minute in comparison with the mass of the lukewarm and the misguided. ...
Humanity is going to need a substantially new way of thinking if it is to survive!" (Albert Einstein)

We can now deduce the most simple science theory of reality - the wave structure of matter in space. By understanding how we and everything around us are interconnected in Space we can then deduce solutions to the fundamental problems of human knowledge in physics, philosophy, metaphysics, theology, education, health, evolution and ecology, politics and society.

This is the profound new way of thinking that Einstein realised, that we exist as spatially extended structures of the universe - the discrete and separate body an illusion. This simply confirms the intuitions of the ancient philosophers and mystics.

Given the current censorship in physics / philosophy of science journals (based on the standard model of particle physics / big bang cosmology) the internet is the best hope for getting new knowledge known to the world. But that depends on you, the people who care about science and society, realise the importance of truth and reality.

It is Easy to Help!

Just click on the Social Network links at top of page, or copy a nice image or quote you like and share it. We have a wonderful collection of knowledge from the greatest minds in human history, so people will appreciate your contributions. In doing this you will help a new generation of scientists see that there is a simple sensible explanation of physical reality (One Substance, One Law) - the source of truth and wisdom, the only cure for the madness of man! Thanks! Geoff Haselhurst (Updated May, 2025)

A new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die, and a new generation grows up that is familiar with it. (Max Planck, 1920)

"All that is necessary for evil to succeed is for good people to do nothing."
(Edmund Burke)

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The Wave Structure of Matter (WSM) One Substance - One Law - One Logic