There was - I think on this very forum - an article linked some time ago. In it, a layman-friendly explanation of why science can prove that telekinesis - defined as moving objects by thought alone - is not possible. The central thesis was that for thought alone to move something, the brain (or another organ responding to brain signals - i.e. thought - somewhere in the human body) must be generating force to actuate the motion. We have several known forces; none of these can the human brain - or, indeed, any organ in the human body - generate in quantities and with the control required to create telekinetic manipulation of objects.
Therefore, if telekinesis is a thing - or at least, so goes the argument - it must be a hitherto unknown force. Here's where the article gets interesting. It makes the claim - believably, though not sufficiently that I can repeat the argument from memory - that such an unknown force would have already been detected if it had sufficient strength-at-range to perform as indicated. More precisely, for it to be strong enough to do what is described, it would have to have influences that we would already have detected. If it is not detectable by our modern science, it must exist on a range that is sufficiently small as to cause it to not act beyond its generating particle.
So, I can buy that. We certainly don't have undisputed documented evidence of TK-users.
However, my engineering brain latched onto this problem of proof-by-negation. It relies on the contradiction of sufficient range and sufficient strength being such that we would have detected it if it had both.
What, however, if there were two forces acting in tandem? One with sufficient range, and one with sufficient strength? It would require that the object being moved actually generate the force, and use it to act upon itself in some way to cause motion. This raises the sticky question of how that's TK, and how it would be at the thought-behest of the human telekinetic.
What if the human brain - or another organ in the body - CAN generate a force of sufficient range, but such insignificant strength that we've yet to figure out how to detect it? Well, if we assume that some very fine sensor in our body or brain can detect said force, that could, first of all, explain telepathy; we send signals back and forth that only our brains can detect!
But what about telekinesis?
For this, I turn your attention to how your arm works. Your brain - possessed of a thought - commands the arm to move. Perhaps to click "reply" beneath this post, so you can tell me I'm full of it (because I am, quite honestly, being quite venturesome into the realm of "what if" in the interest of constructing an argument for something in which I am not an actual believer). When you order your arm and hand to do this, your brain doesn't generate a force that actually causes your finger to be externally manipulated. No, what it does is send an electrical signal to your muscles, which respond to the signal by triggering additional biomechanical forces and contracting in specific ways so as to cause your arm to move the mouse and click the mouse button.
But this example gets even better.
Let's say that the button isn't the "reply" button on this board, but a button that is an interface control for a simple toy crane that will pick up a marble and drop it at the start of a Rube-Goldberg device.
Having clicked the mouse, is the motion of your finger generating force that causes the crane to move? No, again, it is not. It sends still more electric signals racing through your computer, until a command signal exits the port connected to the crane, and the crane's servos and motors generate the forces necessary.
A similar example comes from the R/C car, such as many of us doubtless owned when we were small children. The little kid's brain is not reaching out and manipulating the car, exerting external force on it. It is sending signals to the kid's hands, which are creating force in his muscles of his fingers, and which then manipulate signal-generation switches on the remote control. These, in turn, send radio signals to the car. The radio signals can't move something of the car's mass even one milimeter. However, they induce signals in the car which trigger motors and servos and cause the car to self-propel, drawing power from its battery as needed to do so.
Is the child telekinetic? To ancient peoples, he would certainly seem so. Or at least wizardly, wielding the strange boxy wand to command the strange tiny demonic creature that whirrs along on the ground at his whim.
So, then, what if there is a force our brains (or other organ in our bodies) can generate which is so weak that our technology cannot yet detect it? What if that force interacts with base matter in a particular way, inducing command signals which cause ordered release of a very strong, short-ranged force whose range is, likewise, so small that our technology cannot detect it? We still would need to explain from whence the energy to power the force's movement of the object comes from, but this does neatly side-step the scientific explanation for why humans cannot, supposedly, be telekinetic.
This is, in essence, what my quote asking ukaserex was intended to hint at: I would have, by thinking and thus inducing my fingers to type and send the post, caused a signal to reach ukaserex's brain which, should he have chosen to comply, would have resulted in his arm being raised. Now, we can argue that the fact he has a choice means that I am divorced from the action sufficiently to deny any claim of TK. But...the R/C car has no free will, and it would move as I commanded, were I to be in possession of its remote.