Differential equations you say? You mean the type of equations, that given the state of the system at any time, the states for all other times may be calculated?

No, I mean simply equations that relate functions to their derivatives (of any order): i.e. the mathematical definition of a differential equation. It would perhaps help the discussion if you were aware of some basic mathematical terminology.

And once I again I have to ask the question I posed: what is your definition of determinism lying behind your claim that modern science is deterministic and so eliminates the notion of causality.

Yet you still believe that science is about modeling known causal relations mathematically, and thus miraculously capturing unknown causal and acausal relations, without being aware of what you are doing.

Some science very definitely is involved in modelling supposed causal relations, I never made the claim that all science is about modelling causal relations.

Read Suppes's critique of Russell's position regarding physics not using the notion of causality, before just citing Russell as an authority on the subject. Russell got many things wrong - more things wrong than he got right, by some lights.

No, I mean simply equations that relate functions to their derivatives (of any order): i.e. the mathematical definition of a differential equation. It would perhaps help the discussion if you were aware of some basic mathematical terminology. — MetaphysicsNow

When the ignorant impute ignorance, it is never pretty, is it?

Now, because I am "unaware of basic mathematical terminology", perhaps you could find, using your deep knowledge and expertise in mathematics, one of these "differential equations" pertaining to a physical system, that does not determine the past and the future given initial conditions at any time?

First, differential equations themselves do not determine anything. Second, if your point is that all physical laws are time-symmetric, you are not accounting for the Second Law of Thermodynamics. That allows for some time-symmetric solutions where there are no changes in entropy of a system, but where changes in entropy are concerned, we have irreversibility and, yes, there are differential equations (to be specific, partial differential equations) that are used to model entropy changes. So from those equations you cannot get any results about the past of the system, only regarding the future.

Now that I have met your challenge, how about responding to my question (for the third time of asking): what conception of determinism are you working with that frees it from the notion of causality?

First, differential equations themselves do not determine anything. Second, if your point is that all physical laws are time-symmetric, you are not accounting for the Second Law of Thermodynamics. That allows for some time-symmetric solutions where there are no changes in entropy of a system, but where changes in entropy are concerned, we have irreversibility and, yes, there are differential equations (to be specific, partial differential equations) that are used to model entropy changes. — MetaphysicsNow

So, you cannot find a single case where a physical system, whose time-evolution is determined by laws of motion, expressed in differential equation form, is not set for all times given a set of initial conditions.

And you have the temerity to accuse me of being "unaware of basic mathematical terminology"!

Now, as for the 2nd law, are you seriously suggesting it cannot be used to calculate the entropy in the past?

So, you cannot find a single case where a physical system, whose time-evolution is determined by laws of motion, expressed in differential equation form, is not set for all times given a set of initial conditions.

You asked about differential equations for physical systems in general, not laws of motion in specific, so the example I gave concerned thermodynamics, not dynamics. But even so, on a general point about the use of differential equations given initial conditions, the differential equations will allow you to calculate the future changes of the system, I've not denied that at any point, but they will not allow you to calculate how the system arrived in that state, so you cannot use the laws of motion with just initial conditions to calculate how things were in the past. Why? Amongst other things, for the simple reason that there is no representation within the differential equations of dynamics or thermodynamics of system for the amount of time for which the system has been in that initial state.

Of course, given an initial conditionand a terminating condition, the equations of dynamics and thermodynamics will allow you to get from one to the other in either direction by making appropriate reversals to the time-dependent variables and their derivatives, but that was not the question you asked.

Now, will you please provide us with some precision on what you take determinism to be?

It seems to me that we commonly regard the "non-physical" as being concepts. But concepts are physically heard even as thoughts.

So either everything is physical or nothing is. I lean towards nothing.

Of course, given an initial conditionand a terminating condition, the equations of dynamics and thermodynamics will allow you to get from one to the other in either direction by making appropriate reversals to the time-dependent variables and their derivatives, but that was not the question you asked. — MetaphysicsNow

You are exhibiting the stratospheric level of cluelessness frequently encountered in those who impute ignorance to others.

OK, you win, I'm an idiot. Now can we please have a clear explanation from you concerning what you take determinism to be and how it is freed from any notion of causation?

What cluelessness is MetaphysicsNow manifesting? I was under the impression that time-reversal symmetry in physics was precisely the idea that given intial conditions and terminating conditions, you can get from the latter to the fomer by reversing the time-dependent parameters. I've heard it colloquially explained in terms of there being no physical difference between a film of two billiard balls colliding whether run backwards or forwards. That seems to be what MN is getting at. What's the correct view?

Tom seems to have gone into stealth mode, but I - like you -await with bated breath not only for his explication of time-reversal symmetry but also his explanation of causation-free determinism.

What cluelessness is MetaphysicsNow manifesting? I was under the impression that time-reversal symmetry in physics was precisely the idea that given intial conditions and terminating conditions, you can get from the latter to the fomer by reversing the time-dependent parameters. — jkg20

Nope, you only need initial conditions, which can be given at any time. Differential equations are by their very nature time-symmetric, deterministic. The laws of nature, expressed as differential equations, are of low order, and the most important one is even linear.

So, come out from under your impression into the light.

Initial conditions by themselves don't tell you how things were prior to those conditions, this is the fundamental error you are making. An initial condition at time t involving a ball with constant acceleration a an initial velocity v and an inital spatial location p will determine the forward trajectory of that ball. But without further information about that system before t - and so information not available from the initial conditions - no equations will give you the path it took to reach that initial state, since that initial state is compatible with an indefinite number of previous occurences. For instance, suppose that just up to t the object had a constant acceleration of (a+1) but was, just before t, subjected momentarily to a small decelerating force reducing the acceleration to the constant a. Suppose again that just up to t the object had a constant acceleration of (a+2) but was, just before t, subjected momentarily to a slightly larger decelerating force reducing the acceleration to the constant a. Nothing in the initial conditions, nor the equations of motion you choose to use will allow you to work backward to one or other of those previous states. The application of time-reversal symmetry requires initial conditions and terminating conditions.

Initial conditions by themselves don't tell you how things were prior to those conditions, this is the fundamental error you are making. An initial condition at time t involving a ball with constant acceleration a an initial velocity v and an inital spatial location p will determine the forward trajectory of that ball. — MetaphysicsNow

Oh dear!

The conditions at any time give you the future, and the past. You seem confused what these are. Acceleration is not one of them.

My definition is meta physics.

Something we can freely speculate about until we develop a method for observing the phenomenon.

What are you talking about, your reply makes no sense whatsoever? In dynamics, if your system involves a particle in motion, part of specifying the intial conditions for that system is to specify the particle's acceleration and whether or not it is constant.

Have you ever actually done any physics rather than just talking about it?

What are you talking about, your reply makes no sense whatsoever? In dynamics, if your system involves a particle in motion, part of specifying the intial conditions for that system is to specify the particle's acceleration and whether or not it is constant. — MetaphysicsNow

The "acceleration" is captured by the Hamiltonian, not the initial conditions.

Have you ever actually done any physics rather than just talking about it? — MetaphysicsNow

As a matter of fact, yes.

OK, so you are talking about Langrangian-Hamiltonian mechanics, whereas my example was expressed in the context of Newtonian mechanics. The principle remains the same. Initial conditions themselves using Langrangian-Hamiltonian mechanics will allow you to predict how the system will evolve. Initial conditions themselves plus tools of Langrangian-Hamiltonian mechanics will not allow you to display how those initial conditions arose in the first place - they are not even designed to do that.

Some technical points for Tom and Metaphysics Now to consider:

Equilibrium thermodynamics does not model how entropy changes in time because entropy is by definition a state function that is only defined...at equilibrium. Modern thermodynamics, or non-equilibrium thermodynamics, handles how entropy can change in time, but there the definition of entropy is a bit more controversial.

Differential equations are usually deterministic (ordinary) or stochastic. The Schrodinger equation is an example of a deterministic equation: it tells you the future value of the wavefunction for all time once you have a potential energy source and initial conditions. The Langevin equation is an example of a stochastic differential equation. It looks a bit like this:

Langevin = Some average drift term + Some (usually) Gaussian diffusion term that models uncertainty and interactions

Stochastic equations like this one have either strong or weak solutions. The strong solutions are guaranteed to be unique for a given set of initial conditions, as famously proved by Ito. But the weak solutions only have to satisfy the constraints of a special probability space, so they are not "unique" in the traditional sense we know from ordinary differential equations. This can lead to the amazing result that the same initial conditions do not yield the same final answer using intensive computational algorithms (numerical solutions).

Newton's second law is one formalism that specifies the dynamics of a classical system. It is a second-order differential equation. To solve it you need the position and velocity of a system at some given time t. You do not have to specify acceleration to solve the second law.

I don't mean to suggest anything about what "the state of reality is like" when I say all this. I'm staying out of the metaphysical debate. Just wanted to mention these points for your consideration.

Also I don't know what Tom means when saying that the acceleration is "captured" by the Hamiltonian. The canonical coordinates of the Hamiltonian formalism are position and momentum. Maybe Tom means that the time derivative of momentum in Hamilton's equations reduces to the second law?

The Hamiltonian is just the total energy of the system, good old T + V.

OK, so you are talking about Langrangian-Hamiltonian mechanics, whereas my example was expressed in the context of Newtonian mechanics. — MetaphysicsNow

You think there is a fundamental difference?

In Newtonian mechanics, the "accelleration" is captured by the "Hamiltonian" as well - the forces acting in other words, it is not an initial condition.

Thanks for that Uber. The current bone of contention is the idea of time-reversal symmetry. My claim is that in order to display time-reversal symmetry in physics you need both initial and terminating conditions plus whatever mathematical tools you are using to model the system (Newtonian or Langrangian-Hamiltonian). Tom seems to believe that you need only intial conditions plus the mathematical tools and you can work your way backward or forward willy-nilly.

Also I don't know what Tom means when saying that the acceleration is "captured" by the Hamiltonian. The canonical coordinates of the Hamiltonian formalism are position and momentum. Maybe Tom means that the time derivative of momentum in Hamilton's equations reduces to the second law? — Uber

What time evolves the initial canonical coordinates?

I don't understand that question. The canonical coordinates evolve according to Hamilton's equations.

Newton's second law is one formalism that specifies the dynamics of a classical system. It is a second-order differential equation. To solve it you need the position and velocity of a system at some given time t. You do not have to specify acceleration to solve the second law. — Uber

So acceleration is not one of the initial conditions, and is captured in the Hamiltonian or whatever you like to call the force and potential terms that evolve the state of the system with time.

Acceleration does not explicitly appear in the formalism. The time derivative of momentum (equivalent to the force) is set equal to the partial derivative of the Hamiltonian with respect to position, which reduces to the negative gradient of potential energy.

I have encountered the same difficulties with the non-physical as other respondents. I finally realised that, when I want to use the term non-physical or physical, what I intend is to distinguish the space-time universe (that science describes so ably) from everything else. [Where 'everything else' is more or less all-inclusive, and includes God and religion, politics, science (the discipline, not its subject matter), philosophy, art, design, music, and so on.] I don't actually have a term for this, but I know what I mean when I think about it. As for communicating these ideas (clearly) to others ... I'm still working on that. :chin: :smile: I'm open to helpful suggestions? :up:

@tom

You think there is a fundamental difference?

Between Newtonian and Hamiltonian mechanics? Depends what you mean by fundamental. But in any case there is a difference in the tools they provide to solve problems. But all of that is irrelevant to the dispute about time-reversal symmetry.

The purport of time-reversal symmetry is that it does not matter from which of two given time-separated physical states you begin, you can calculate the transition to the other state using precisely the same physical laws because those laws are invariant under the reversal of sign of the temporal parameters. This requires that the two states are distinct and indentifiable in such a way that the necessary values to plug into the physical laws are available. One of the states you can define as "initial conditions" - it does not matter which - the other "terminating conditions". Time-reversal symmetry says nothing about what you can predict about the past just given a physical description of a system at one given moment.