How do you figure things out?  22 ways 
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Physics  
Physics John Harland and I talk about physics. How might we think of it in terms of "ways of figuring things out" and my overview of that? John and I were graduate students at UCSD in the math department where we received our Ph.D's. I have a B.A. in Physics and I think John does, too, but he certainly knows and thinks a lot more about physics than I do. I share my notes based on my understanding of ideas that John sparked or stated and I tried to make sense of in my system. 849  
Measurement  
Measurement We can always start fresh with a new measurement. Each measurement assumes a partial view, an interest in some part of the system. We don't need a complete description, but rather we tease out whatever part of reality that we are interested, even though it is dubious in the big picture, yet our point of view (say, particle or wave) can be successful, even though incomplete. Yet therefore we need to keep working with independent measurements. Analogously, in math we can start fresh with a new piece of paper, or in life we can give a person a new chance. 850  
Particle point of view  
Particle point of view Our measurement can take place within the frame of measurement. We have a natural frame of reference, for example, the center of mass. That center of mass can then be considered as balancing different masses, and integrating a system of masses, and ultimately defining a vector space. This is a static, spatial, nontemporal point of view. Every state has a location. We can speak of the state of a system. Analogously, in math we have a blank sheet with a natural frame, a center, a balance around that center, a polynomial algebra of constructions, and ultimately, a vector space where a basis makes explicit that every point can be the center. And in life, we can discard the unessential, presume only God, allow for both self and others, find harmony amongst our interests, and create a space for good Spirit. 851  
Wave point of view  
Wave point of view Our measurement can take place outside of our frames of measurement and thus link several such frames. This is a dynamic point of view where there is no distinction between the future and the past so that all is reversible. We can think of the wave in terms of where it starts and where it ends. It includes all paths between these two points. This is a deterministic, nonspatial point of view, which establishes time, an ideal continuum that is beyond the frames but thus relevant for us. Analogously, in math we may have a sequence of sheets, as with mathematical induction, some of which may be of ultimate importance, as with the extreme principle, thus allowing for boxing in with greatest lower bounds and least upper bounds, leading to limits that may transcend, go beyond what we can account for. Or in life, we can be open to care about everything, then care about our minds by which we care, then come up against our personal limits, then allow for an ideal (such as Jesus) that transcends our limits. 848  
Variational principle  
Variational principle Wikipedia: A variational principle is a scientific principle used within the calculus of variations, which develops general methods for finding functions which minimize or maximize the value of quantities that depends upon those functions. For example, to answer this question: "What is the shape of a chain suspended at both ends?" we can use the variational principle that the shape must minimize the gravitational potential energy. According to Cornelius Lanczos, any physical law which can be expressed as a variational principle describes an expression which is selfadjoint. These expressions are also called Hermitian. Such an expression describes an invariant under a Hermitian transformation. Felix Klein's Erlangen program attempted to identify such invariants under a group of transformations. In what is referred to in physics as Noether's theorem, the PoincarÃƒÆ’Ã‚Â© group of transformations (what is now called a gauge group) for general relativity defines symmetries under a group of transformations which depend on a variational principle, or action principle. See: History of variational principles in physics and Brachistochrone, the curve of fastest descent907 Fermat's principle Wikipedia: In optics, Fermat's principle or the principle of least time is the principle that the path taken between two points by a ray of light is the path that can be traversed in the least time. This principle is sometimes taken as the definition of a ray of light. However, this version of the principle is not general; a more modern statement of the principle is that rays of light traverse the path of stationary optical length. Fermat's principle can be used to describe the properties of light rays reflected off mirrors, refracted through different media, or undergoing total internal reflection. It follows mathematically from Huygens' principle (at the limit of small wavelength), and can be used to derive Snell's law of refraction and the law of reflection.908 Gauss' principle of least constraint Wikipedia: The principle of least constraint is a least squares principle stating that the true motion of a mechanical system of N masses is the minimum of the quantity above for all trajectories satisfying any imposed constraints, where mk, rk and Fk represent the mass, position and applied forces of the kth mass. Gauss' principle is equivalent to D'Alembert's principle. The principle of least constraint is qualitatively similar to Hamilton's principle, which states that the true path taken by a mechanical system is an extremum of the action. However, Gauss' principle is a true (local) minimal principle, whereas the other is an extremal principle.904 Hertz's principle of least curvature Wikipedia: Hertz's principle of least curvature is a special case of Gauss' principle, restricted by the two conditions that there be no applied forces and that all masses are identical.905 Principle of least action Wikipedia: In physics, the principle of least action  or, more accurately, the principle of stationary action  is a variational principle that, when applied to the action of a mechanical system, can be used to obtain the equations of motion for that system. The principle led to the development of the Lagrangian and Hamiltonian formulations of classical mechanics. ... Maupertuis felt that "Nature is thrifty in all its actions" ... the action principle is not localized to a point; rather, it involves integrals over an interval of time and (for fields) an extended region of space. Moreover, in the usual formulation of classical action principles, the initial and final states of the system are fixed, e.g., Given that the particle begins at position x1 at time t1 and ends at position x2 at time t2, the physical trajectory that connects these two endpoints is an extremum of the action integral. In particular, the fixing of the final state appears to give the action principle a teleological character which has been controversial historically.903  
Scientific method  
Scientific method We design experiments that link together, tangle together the two incomplete outlooks of space and time, single frame and multiple frames, particle and wave, static and dynamic, free and deterministic. This is because each experiment presumes an experimenter and thus takes place both within a frame of measurement and beyond it. Each experiment includes a hypothesis, an experimental test, and an appraisal of the results. Analogously, in math, given a constraint, we extend its domain to include a new domain, we stitch them together by presuming continuity, and we relate the two applications by superimposing them, yielding a more general constraint. In life, we take a stand, follow through and reflect. 852 Physics experiments Wikipedia documents more than 200 physics experiments. The experiment page gives examples of how the scientific method is applied. There is also a page listing key physics experiments.877  
Hypothesis  
Hypothesis Wikipedia: A hypothesis is a proposed explanation for a phenomenon. The term derives from the Greek, hypotithenai meaning "to put under" or "to suppose." For a hypothesis to be put forward as a scientific hypothesis, the scientific method requires that one can test it. Scientists generally base scientific hypotheses on previous observations that cannot satisfactorily be explained with the available scientific theories.913 Mapping observables and observations Edward Cherlin, 2011.04.24: I like your cycle of scientific method: take a stand (hypothesize), follow through (experiment), reflect (conclude), although I find that there is more to it. It has been pointed out that a hypothesis must include a model (usually mathematical) and a mapping between parts of the model (observables) and observations, including experiments. 911  
Experimental design  
Experimental design Wikipedia: In general usage, design of experiments (DOE) or experimental design is the design of any informationgathering exercises where variation is present, whether under the full control of the experimenter or not.914 Design experiments to rule models in or out Edward Cherlin, 2011.04.24: But that is not enough. We must also think of other possible models, and design experiments to rule them in or out, and we must think of every possible experiment that could invalidate our model. This is the great service that Einstein performed for Quantum Mechanics, because he disliked it so much. Every time he thought he had found a contradiction or something nonsensical in the math, the lab boys verified that it really worked that way in experiments. 912  
Observe  
Observe 915 Extend our senses with improved scientific instruments Edward Cherlin, 2011.04.24: We know that our models are reasonably complete and accurate at best in the areas we have been able to observe, and that every new addition to our senses in improved scientific instruments, going back to Galileo's first telescope, reveals surprises like the mountains of the moon, the constancy of the speed of light (interferometers) or neutrino oscillation (simple but quite large neutrino detectors).916  
Sets of objects exist  
Sets of objects exist We can fuse the particle and wave points of view to work with a partial reality. For example, we can talk about a banana, an apple or an orange as well defined objects that mean something more than a random assemblage of half of the atoms in a banana with half of the atoms in the apple. We are then no longer talking about the symmetry of the universe. John says: A symmetry group commutes with the underlying symmetry of a particular phenomenon, its spacial symmetry, as the set of possible transformations, possible futures. Previously, we worked with the entire universe, and if we translated it abstractly in a symmetry group and then ran things forward in the translated frame, it was exactly isomorphic to what it would have been if we had not translated it. But now, as we want to compartmentalize the universe, then the price to pay is that we are not translating by the full symmetry group, but only by some part of it. This is analogous to having a tensor product and considering only one component, so that we have partial symmetry. We are going to treat one part of the universe as a compartment. This gives the reality to the symmetry group because otherwise it couldn't be measured. Andrius: This compartmentalization is also what allows us to define entropy and the oneway direction of time, which says that states drift away from deliberateness, which is expressed by the compartmentalization. Compartmentalization also indicates the philosophical gaps or boundaries that allow for measurement to take place, allow for objectivity, the separation of the observer and the observed. Analogously, in math we have symmetry groups, and in life we have meanings, the essence of what we wish to say, which me take to be absolute, in cases where we have fundamental agreement. 847  
Conservation  
Conservation 998 Noether's Theorem Wikipedia: Noether's (first) theorem states that any differentiable symmetry of the action of a physical system has a corresponding conservation law. ... The action of a physical system is the integral over time of a Lagrangian function (which may or may not be an integral over space of a Lagrangian density function), from which the system's behavior can be determined by the principle of least action. Noether's theorem has become a fundamental tool of modern theoretical physics and the calculus of variations. ... For illustration, if a physical system behaves the same regardless of how it is oriented in space, its Lagrangian is rotationally symmetric; from this symmetry, Noether's theorem shows the angular momentum of the system must be conserved. ... Noether's theorem is important, both because of the insight it gives into conservation laws, and also as a practical calculational tool. It allows researchers to determine the conserved quantities from the observed symmetries of a physical system. Conversely, it allows researchers to consider whole classes of hypothetical Lagrangians to describe a physical system. For illustration, suppose that a new field is discovered that conserves a quantity X. Using Noether's theorem, the types of Lagrangians that conserve X because of a continuous symmetry can be determined, and then their fitness judged by other criteria.981 Conservation law Wikipedia: Noether's (first) theorem states that any differentiable symmetry of the action of a physical system has a corresponding conservation law. ... The action of a physical system is the integral over time of a Lagrangian function (which may or may not be an integral over space of a Lagrangian density function), from which the system's behavior can be determined by the principle of least action. Noether's theorem has become a fundamental tool of modern theoretical physics and the calculus of variations. ... For illustration, if a physical system behaves the same regardless of how it is oriented in space, its Lagrangian is rotationally symmetric; from this symmetry, Noether's theorem shows the angular momentum of the system must be conserved. ... Noether's theorem is important, both because of the insight it gives into conservation laws, and also as a practical calculational tool. It allows researchers to determine the conserved quantities from the observed symmetries of a physical system. Conversely, it allows researchers to consider whole classes of hypothetical Lagrangians to describe a physical system. For illustration, suppose that a new field is discovered that conserves a quantity X. Using Noether's theorem, the types of Lagrangians that conserve X because of a continuous symmetry can be determined, and then their fitness judged by other criteria.982 Conservation of angular momentum Wikipedia: In a closed system angular momentum is constant. This conservation law mathematically follows from continuous directional symmetry of space (no direction in space is any different from any other direction). If a planet is found to rotate slower than expected, then astronomers suspect that the planet is accompanied by a satellite, because the total angular momentum is shared between the planet and its satellite in order to be conserved. The conservation of angular momentum is used extensively in analyzing what is called central force motion. If the net force on some body is directed always toward some fixed point, the center, then there is no torque on the body with respect to the center, and so the angular momentum of the body about the center is constant. 985 Conservation of color charge Wikipedia: In particle physics, color charge is a property of quarks and gluons that is related to the particles' strong interactions in the theory of quantum chromodynamics (QCD). ... Color charge is conserved, but the bookkeeping involved in this is more complicated than just adding up the charges, as is done in quantum electrodynamics.987 Conservation of electric charge Wikipedia: In physics, charge conservation is the principle that electric charge can neither be created nor destroyed. The quantity of electric charge, the amount of positive charge minus the amount of negative charge in the universe, is always conserved. Charge conservation is a physical law that states that the change in the amount of electric charge in any volume of space is exactly equal to the amount of charge flowing into the volume minus the amount of charge flowing out of the volume. In essence, charge conservation is an accounting relationship between the amount of charge in a region and the flow of charge into and out of that region.986 Conservation of linear momentum Wikipedia: The law of conservation of linear momentum is a fundamental law of nature, and it states that if no external force acts on a closed system of objects the momentum of the closed system remains constant. One of the consequences of this is that the center of mass of any system of objects will always continue with the same velocity unless acted on by a force from outside the system. Conservation of momentum is a mathematical consequence of the homogeneity (shift symmetry) of space (position in space is the canonical conjugate quantity to momentum). So, momentum conservation can be philosophically stated as "nothing depends on location per se".984 Conservation of massenergy Wikipedia: In physics, massenergy equivalence is the concept that the mass of a body is a measure of its energy content. In this concept the total internal energy E of a body at rest is equal to the product of its rest mass m and a suitable conversion factor to transform from units of mass to units of energy. If the body is not stationary relative to the observer then account must be made for relativistic effects where m is given by the relativistic mass and E the relativistic energy of the body.983 Conservation of probability density Wikipedia: In quantum mechanics, the probability current (sometimes called probability flux) is a concept describing the flow of probability density. In particular, if one pictures the probability density as an inhomogeneous fluid, then the probability current is the rate of flow of this fluid (the density times the velocity). ... This is the conservation law for probability in quantum mechanics.... 989 Conservation of weak isospin Wikipedia: In particle physics, weak isospin is a quantum number relating to the weak interaction, and parallels the idea of isospin under the strong interaction. ... The weak isospin conservation law relates the conservation of T3; all weak interactions must preserve T3. It is also conserved by the other interactions and is therefore a conserved quantity in general.988 CPT symmetry Wikipedia: CPT symmetry is a fundamental symmetry of physical laws under transformations that involve the inversions of charge, parity, and time simultaneously. ... The CPT theorem requires the preservation of CPT symmetry by all physical phenomena. It assumes the correctness of quantum laws and Lorentz invariance. Specifically, the CPT theorem states that any Lorentz invariant local quantum field theory with a Hermitian Hamiltonian must have CPT symmetry.990 Lorentz symmetry Wikipedia: In standard physics, Lorentz symmetry is "the feature of nature that says experimental results are independent of the orientation or the boost velocity of the laboratory through space".991  
Experiments and Theory  
Experiments and Theory Experiments (specific instances) and theory (general laws) are related as level and metalevel. There is a dualism. But, actually, they are not qualitatively different. For an experiment is never a single instance, but always a set of instances, for it must be reproducible. In that sense, every experiment has a generality, just as a theory does. These two levels can be conflated, which is how we view Reality, where the facts and the laws coincide. Or the levels can be distinct to various degrees, and completely distinct when the facts are considered to be applications of the rules. Andrius: There are four possible levels (Whether, What, How, Why) for relating facts and rules, and there are six pairs of possible levels, with the wider level reserved for the rules (the imagined observer) and the narrower level reserved for the facts (the imagined observed). Analogously, in Math we have the mathematical structures that describe (on paper) our problem, and we have the mathematical structures that describe how our minds are solving the problem. The two are conflated as Truth. They are distinguished as Model, Implication and Variable. There are six kinds of variables. In life, we have four ways of distinguishing the truths of the heart and the world, given by Whether, What, How, Why we know what we know, and there are six ways that the two truths may be related. 853  
Symmetry breaking  
Symmetry breaking 1029 Explicit symmetry breaking Wikipedia: Explicit symmetry breaking indicates a situation where the dynamical equations are not manifestly invariant under the symmetry group considered. This means, in the Lagrangian (Hamiltonian) formulation, that the Lagrangian (Hamiltonian) of the system contains one or more terms explicitly breaking the symmetry. Such terms can have different origins:
Spontaneous symmetry breaking Wikipedia: Spontaneous symmetry breaking is the process by which a system described in a theoretically symmetrical way ends up in an apparently asymmetric state. For spontaneous symmetry breaking to occur, there must be a system in which there are several equally likely outcomes. The system as a whole is therefore symmetric with respect to these outcomes (if we consider any two outcomes, the probability is the same). However, if the system is sampled (i.e. if the system is actually used or interacted with in any way), a specific outcome must occur. Though we know the system as a whole is symmetric, we also see that it is never encountered with this symmetry, only in one specific state. Because one of the outcomes is always found with probability 1, and the others with probability 0, they are no longer symmetric. Hence, the symmetry is said to be spontaneously broken in that theory. Nevertheless, the fact that each outcome is equally likely is a reflection of the underlying symmetry, which is thus often dubbed "hidden symmetry", and has crucial formal consequences, such as the presence of NambuGoldstone bosons. When a theory is symmetric with respect to a symmetry group, but requires that one element of the group is distinct, then spontaneous symmetry breaking has occurred. The theory must not dictate which member is distinct, only that one is. 1027 A ball on top of a hill Wikipedia: A common example to help explain this phenomenon is a ball sitting on top of a hill. This ball is in a completely symmetric state. However, its state is unstable: the slightest perturbing force will cause the ball to roll down the hill in some particular direction. At that point, symmetry has been broken, because the direction in which the ball rolled has a visible feature that distinguishes it from all other directions. The "choice" of direction is immaterial, however, as any other direction would do, i.e. the system is still bearing traces of the symmetry of the hill, albeit now somewhat less apparent.1028 Symmetry breaking Wikipedia: Symmetry breaking in physics describes a phenomenon where (infinitesimally) small fluctuations acting on a system which is crossing a critical point decide the system's fate, by determining which branch of a bifurcation is taken. To an outside observer unaware of the fluctuations (or "noise"), the choice will appear arbitrary. This process is called symmetry "breaking", because such transitions usually bring the system from a disorderly state into one of two definite states. Since disorder is more symmetric, in the sense that small variations to it don't change its overall appearance, the symmetry gets "broken". Symmetry breaking is supposed to play a major role in pattern formation.1025 Phase transition Wikipedia: A phase transition is the transformation of a thermodynamic system from one phase or state of matter to another. A phase of a thermodynamic system and the states of matter have uniform physical properties. During a phase transition of a given medium certain properties of the medium change, often discontinuously, as a result of some external condition, such as temperature, pressure, and others. For example, a liquid may become gas upon heating to the boiling point, resulting in an abrupt change in volume. The measurement of the external conditions at which the transformation occurs is termed the phase transition point. The term is most commonly used to describe transitions between solid, liquid and gaseous states of matter, in rare cases including plasma.1030  
Forces  
Forces Forces are expressions of causality, of the relationship between before and after. They allow for a system to break down into subsystems. Rules are applied in six different ways to link states before and after an event. In math, these are the kinds of subsystems that Implication forms. In life, these are the ways that we visualize. 854  
Identity  
Identity John says that the whole concept of identity is very flexible in physics, even shaky. What is a storm? Where does it go when it no longer exists? What is the reality of the kludge? You can't know where things come and go in physics. Once you write about it you are not talking about physics in the big picture. This is where there is slack and wiggle room. Identity is what allows for eternal nature. We acknowledge the storm and, though it comes and goes, yet its eternal nature jumps over into us. This is eternal life. In math, there is Context, which can change the meaning entirely, as when 10+4 turns out to be 2'oclock on a clock. In life, if we obey God, then we can imagine God's point of view, and that opens up incredible freedom. 855  
Other ways: Physics  
Other ways: Physics 862 Classifying 1031 Ehrenfest classification of phase transitions Wikipedia: Paul Ehrenfest classified phase transitions based on the behavior of the thermodynamic free energy as a function of other thermodynamic variables. Under this scheme, phase transitions were labeled by the lowest derivative of the free energy that is discontinuous at the transition. Firstorder phase transitions exhibit a discontinuity in the first derivative of the free energy with respect to some thermodynamic variable.[1] The various solid/liquid/gas transitions are classified as firstorder transitions because they involve a discontinuous change in density, which is the first derivative of the free energy with respect to chemical potential. Secondorder phase transitions are continuous in the first derivative (the order parameter, which is the first derivative of the free energy with respect to the external field, is continuous across the transition) but exhibit discontinuity in a second derivative of the free energy.[1] These include the ferromagnetic phase transition in materials such as iron, where the magnetization, which is the first derivative of the free energy with the applied magnetic field strength, increases continuously from zero as the temperature is lowered below the Curie temperature. The magnetic susceptibility, the second derivative of the free energy with the field, changes discontinuously. Under the Ehrenfest classification scheme, there could in principle be third, fourth, and higherorder phase transitions.1032 Modern classification of phase transitions Wikipedia: In the modern classification scheme, phase transitions are divided into two broad categories, named similarly to the Ehrenfest classes:
