While dogs understand the concept of time, they don't see it like humans do. Instead of thinking in abstract terms and working off specific memories, they relate the passage of time only to their present. In other words, they aren't oblivious, but they don't wax nostalgic about the past, either.
Humans and Time
People have what scientists call an "episodic memory." This means our understanding of time is based on specific memories of the past, cognizance of the present and anticipation of the future. We think of time as a specific set of periods and events, which we categorize in our brains like home movies. It's how we hold onto memories of major events, like weddings, and develop short and long-term routines, like Christmas traditions or weekly date nights.
How Long Since...
A dog's understanding of time is more grounded in concrete terms, specifically ones that relate to the present. The easiest way to think of this is in "how long since" terms. For example, your dog has a basic understanding of how long since you left the house this morning, how long since you fed him last or how long since his last walk. He doesn't track time like humans do, but he can tell if you left just a few minutes ago or a few hours ago, and it affects his mood.
Learning, Not Remembering
Because dogs don't have episodic memory like humans, they don't think of time in concrete terms -- but that doesn't mean they don't remember at all. Your dog's memory allows him to learn things without remembering why he knows them. For example, if your dog is attacked and hurt by another dog, he won't necessarily remember the event later, but he may experience fear when he sees another dog of that breed. He may not remember his last trip to the vet's office, but the scents and sounds of the waiting room will trigger emotions.
In the Moment
His understanding of time -- and lack thereof -- both allow your dog to adapt and protect himself. He's programmed to live in the moment, forever cognizant of his surroundings, his needs and his potential threats. In theory, a dog doesn't need to remember the way he learned a lesson -- only the lesson itself. This understanding of time and memory allows him to learn from the past and be ready for the future without ever explicitly thinking about either one.
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