Fermat's Last Theorem The 1670 edition of Diophantus' Arithmetica includes Fermat's commentary, particularly his "Last Theorem" (Observatio Domini Petri de Fermat). In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two. This theorem was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of Arithmetica where he claimed he had a proof that was too large to fit in the margin. The first successful proof was released in 1994 by Andrew Wiles, and formally published in 1995, after 358 years of effort by mathematicians. The unsolved problem stimulated the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th century.
Level of consciousness (Esotericism) Consciousness is a loosely defined concept that addresses the human awareness of both internal and external stimuli. This can refer to spiritual recognition, psychological understanding, medically altered states, or more modern-day concepts of life purpose, satisfaction, and self-actualization. Most theories map consciousness in a series of levels, some stages of which are more continuous or complex than others. Movement between stages is often bidirectional depending on internal and external conditions, with each mental ascension precipitating a change in reactivity. In the most basic sense, this alteration might lead to a reduced responsiveness as seen in anesthesiology; more abstract facets of tiered consciousness describe characteristics of profoundness, insight, perception, or understanding. Pyramid of Kulkucan (found at the center of the Chichen Itza)
Mixing (physics) In physics, a dynamical system is said to be mixing if the phase space of the system becomes strongly intertwined, according to at least one of several mathematical definitions. For example, a measure-preserving transformation T is said to be strong mixing if Mixing in a ball of colored putty after consecutive iterations of the Smale horseshoe map (i.e. squashing and folding in two) Quantum mechanics Solution to Schrödinger's equation for the hydrogen atom at different energy levels. The brighter areas represent a higher probability of finding an electron. Quantum mechanics gradually arose from Max Planck's solution in 1900 to the black-body radiation problem (reported 1859) and Albert Einstein's 1905 paper which offered a quantum-based theory to explain the photoelectric effect (reported 1887). Early quantum theory was profoundly reconceived in the mid-1920s.
Planck scale In particle physics and physical cosmology, the Planck scale (named after Max Planck) is an energy scale around 1.22 × 1019 GeV (which corresponds by the mass–energy equivalence to the Planck mass 2.17645 × 10−8 kg) at which quantum effects of gravity become strong. At this scale, present descriptions and theories of sub-atomic particle interactions in terms of quantum field theory break down and become inadequate, due to the impact of the apparent non-renormalizability of gravity within current theories. At the Planck scale, the strength of gravity is expected to become comparable with the other forces, and it is theorized that all the fundamental forces are unified at that scale, but the exact mechanism of this unification remains unknown.
Philosophy of mind A phrenological mapping of the brain – phrenology was among the first attempts to correlate mental functions with specific parts of the brain Philosophy of mind is a branch of philosophy that studies the nature of the mind, mental events, mental functions, mental properties, consciousness, and their relationship to the physical body, particularly the brain. The mind–body problem, i.e. the relationship of the mind to the body, is commonly seen as one key issue in philosophy of mind, although there are other issues concerning the nature of the mind that do not involve its relation to the physical body, such as how consciousness is possible and the nature of particular mental states.
Neural correlates of consciousness Figure 1: The Neuronal Correlates of Consciousness (NCC) are the minimal set of neural events and structures – here synchronized action potentials in neocortical pyramidal neurons – sufficient for a specific conscious percept or a conscious (explicit) memory. From Koch (2004). The Neural Correlates of Consciousness (NCC) can be defined as the minimal neuronal mechanisms jointly sufficient for any one specific conscious percept (Crick & Koch 1990). The Infinity of Quantum Physics Quantum Mechanics the Infinity Photon Electron Couplings creating sets of infinities of Fractal Self-Similarities Holographic principle In a larger sense, the theory suggests that the entire universe can be seen as a two-dimensional information structure "painted" on the cosmological horizon[clarification needed], such that the three dimensions we observe are an effective description only at macroscopic scales and at low energies. Cosmological holography has not been made mathematically precise, partly because the particle horizon has a finite area and grows with time. The holographic principle was inspired by black hole thermodynamics, which conjectures that the maximal entropy in any region scales with the radius squared, and not cubed as might be expected. In the case of a black hole, the insight was that the informational content of all the objects that have fallen into the hole might be entirely contained in surface fluctuations of the event horizon.
Reductionism Descartes held that non-human animals could be reductively explained as automata — De homine, 1662. Reductionism strongly reflects a certain perspective on causality. In a reductionist framework, the phenomena that can be explained completely in terms of relations between other more fundamental phenomena, are called epiphenomena. Often there is an implication that the epiphenomenon exerts no causal agency on the fundamental phenomena that explain it. Reductionism does not preclude the existence of what might be called emergent phenomena, but it does imply the ability to understand those phenomena completely in terms of the processes from which they are composed. This reductionist understanding is very different from that usually implied by the term 'emergence', which typically intends that what emerges is more than the sum of the processes from which it emerges.
Our Conscious Mind Could Be An Electromagnetic Field Are our thoughts made of the distributed kind of electromagnetic field that permeates space and carries the broadcast signal to the TV or radio. Professor Johnjoe McFadden from the School of Biomedical and Life Sciences at the University of Surrey in the UK believes our conscious mind could be an electromagnetic field. “The theory solves many previously intractable problems of consciousness and could have profound implications for our concepts of mind, free will, spirituality, the design of artificial intelligence, and even life and death,” he said. Most people consider "mind" to be all the conscious things that we are aware of. But much, if not most, mental activity goes on without awareness. Actions such as walking, changing gear in your car or peddling a bicycle can become as automatic as breathing.
8 shocking things we learned from Stephen Hawking's book From the idea that our universe is one among many, to the revelation that mathematician Pythagoras didn't actually invent the Pythagorean theorem, here are eight shocking things we learned from reading physicist Stephen Hawking's new book, "The Grand Design," written with fellow physicist Leonard Mlodinow of Caltech. The book, covering major questions about the nature and origin of the universe, was released Sept. 7 by its publisher, Bantam. 1. The past is possibility According to Hawking and Mlodinow, one consequence of the theory of quantum mechanics is that events in the past that were not directly observed did not happen in a definite way.
Matter wave The de Broglie relations redirect here. In quantum mechanics, the concept of matter waves or de Broglie waves /dəˈbrɔɪ/ reflects the wave–particle duality of matter. The theory was proposed by Louis de Broglie in 1924 in his PhD thesis. The de Broglie relations show that the wavelength is inversely proportional to the momentum of a particle and is also called de Broglie wavelength. Also the frequency of matter waves, as deduced by de Broglie, is directly proportional to the total energy E (sum of its rest energy and the kinetic energy) of a particle. Historical context At the end of the 19th century, light was thought to consist of waves of electromagnetic fields which propagated according to Maxwell’s equations, while matter was thought to consist of localized particles (See history of wave and particle viewpoints).